244 papers from 2020, out of 2,678 papers on Ricci flow, Ricci solitons, Perelman's methods and the Poincaré conjecture, plus the geometric flows that share their tools — mean curvature flow, curve shortening, Yamabe flow — marked Wider flows. Click a title for the paper's page, with its abstract, PDF and HTML versions and connected papers, or any blue term to see everything on that topic.
In this work we consider the global existence of volume-preserving crystallinecurvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address global existence and regularity of the flow for smooth anisotropies. For the non-smooth case we establish global existence results for the types of anisotropies known to be globally well-posed.
In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify n-dimensional (n≥5) complete D-flat gradient steady Ricci solitons. More precisely, we prove that any n-dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well.
In this paper, we study the existence of complete Yamabe metric with zero scalar curvature on an n-dimensional complete Riemannian manifold (M,g0), n≥3. Under suitable conditions about the initial metric, we show that there is a global fine solution to the Yamabe flow. The interesting point here is that we have no curvature assumption about the initial metric. We show that on an n-dimensional complete Riemannian manifold (M,g0) with non-negative Ricci curvature, n≥3, the Yamabe flow enjoys the local L1-stability property from the view-point of the porous media equation. Complete Yamabe metrics with zero scalar curvature on an n-dimensional Riemannian model space are also discussed.
In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensionalLorentzianLie groups with some product structure.
Advection and mean curvature flow is used as a model of bone microarchitecture adaptation. It is an equivalent geometric flow to prescribed mean curvature flow with an additional rate term. In order to validate numerical methods for simulating this flow and developing an inverse solver, a closed-form solution for advection and mean curvature flow of a sphere is derived.
We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.
In this paper, we prove that the translating solitons of the mean curvature flow in R4 which arise as blow up limit of embedded, mean convex mean curvature flow must have SO(2) symmetry.
In this paper, we prove that any mean curvature flowtranslatorΣ2⊂R3 with finite total curvature and one end must be a plane. We also prove that if the translator Σ has multiple ends, they are asymptotic to a plane Π containing the direction of translation and can be written as graphs over Π. Finally, we determine that the ends of Σ are strongly asymptotic to Π and obtain quantitative estimates for their asymptotic behavior.
Motivated by Müller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under the generalized Ricci flow.
In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.
In this article we have showed that a gradient ρ-Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient ρ-Einstein soliton with finite weighted Dirichlet integral and certain restriction on Ricci curvature must be of constant scalar curvature and steady Ricci flat. Finally, we have proved that a non-shrinking or non-expanding gradient traceless Ricci soliton possessing some conditions must be steady.
In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.
In this paper we study rotational surfaces in the space H2×R whose mean curvature is given as a prescribed function of their angle function. These surfaces generalize, among others, the ones of constant mean curvature and the translating solitons of the mean curvature flow. Using a phase plane analysis we construct entire rotational graphs, catenoid-type surfaces, and exhibit a classification result when the prescribed function is linear.
In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the L-stability operator and discuss the L-stability of some special self-expanders. Besides, we prove that the surfaces Γ×R with the product metric are the only complete self-expander surfaces immersed in R3 with constant scalar curvature, where Γ is a complete self-expander curve (properly) immersed in R2.
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open curves we propose natural boundary conditions that respect the appropriate gradient flow structure. Based on suitable weak formulations we introduce finite element approximations using piecewise linear elements. For some of the schemes a stability result can be shown. The derived schemes can be employed in very different contexts. For example, we apply the schemes to the Angenent metric in order to numerically compute rotationally symmetricself-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes to compute geodesics that are relevant for optimal interface profiles in multi-component phase field models.
We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the L2 norm of the anisotropic curvature blows up.
We classify closed convex α-curve shortening flows for sub-affine-critical powers α≤31. In addition, we show that closed convex smooth finite entropy α-curve shortening flows with 31<α is a shrinking circle. After normalization, the ancient flows satisfying the above conditions converge exponentially fast to smooth closed convex shrinkers at the backward infinity. In particular, when α=k2−11 with 3≤k∈N, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.
In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete 3-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient – a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.
We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space Rn of dimension at least 2. Our main results states that tangential balls in Rn under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in Rn for a constant forcing term as finite unions of equisized balls with mutually positive distance.
We find the regime of our recently constructed topological nonrelativistic quantum gravity, in which Perelman's Ricci flow equations on Riemannian manifolds appear precisely as the localization equations in the path integral. In this mapping between physics and mathematics, the role of Perelman's dilaton is played by our lapse function. Perelman's local fixed volume condition emerges dynamically as the λ parameter in our kinetic term approaches λ→−∞. The DeTurck trick that decouples the metric flow from the dilaton flow is simply a gauge-fixing condition for the gauge symmetry of spatial diffeomorphisms. We show how Perelman's F and W entropy functionals are related to our superpotential. We explain the origin of Perelman's τ function, which appears in the W entropy functional for shrinking solitons, as the Goldstone mode associated with time translations and spatial rescalings: In fact, in our quantum gravity, Perelman's τ turns out to play the role of a dilaton for anisotropic scale transformations. The map between Perelman's flow and the localization equations in our topological quantum gravity requires an interesting redefinition of fields, which includes a reframing of the metric. With this embedding of Perelman's equations into topological quantum gravity, a wealth of mathematical results on the Ricci flow can now be imported into physics and reformulated in the language of quantum field theory.
In short geometrization conjecture of W.\,Thurston (finally proved by G. Perelman) says that any oriented 3-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal paper (1991) M.\,W.\,Davis and T.\,Januszkiewicz introduced a wide class of n-dimensional manifolds – small covers over simple n-polytopes. We give a complete answer to the following problem: to build an explicit canonical decomposition for any orientable 3-manifold defined by a vector-colouring of a simple 3-polytope, in particular for a small cover. The proof is based on analysis of results in this direction obtained before by different authors.
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem.
We study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern-Ricci-flat metric in the ∂∂ˉ-class of the initial metric.
In this article we prove an unknottedness result for self shrinkers in R3 with multiple asymptotically conical ends which bound a handlebody in a natural sense, using the mean curvature flow. As a corollary of this and previous work, asymptotically conical self shrinkers with two ends are unknotted.
It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension n−3 with n≥4, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.
We construct Lagrangian translating solitons by desingularizing the intersection points between Lagrangian Grim Reaper cylinders with the same phase using special Lagrangian Lawlor necks. The resulting Lagrangian translating solitons could have arbitrarily many ends and non-contractible loops.
In recent work (Pure Appl. Anal. 2 (2020), 397-426), the first named author and J. Zhang found a connection between the regularity theory of optimal transport and the curvature of Kähler manifolds. In particular, we showed that the MTW tensor for a cost function c(x,y)=Ψ(x−y) can be understood as the anti-bisectional curvature of an associated Kähler metric defined on a tube domain. Here, the anti-bisectional curvature is defined as R(X,Y,X,Y) where X and Y are polarized (1,0) vectors and R is the curvature tensor. The correspondence between the anti-bisectional curvature and the MTW tensor provides a meaningful sense in which the anti-bisectional curvature can have a sign (i.e., be positive or negative). In this paper, we study the behavior of the anti-bisectional curvature under Kähler-Ricci flow. We find that non-positive anti-bisectional curvature is preserved under the flow. In complex dimension two, we also show that non-negative orthogonal anti-bisectional curvature (i.e., the MTW(0) condition) is preserved under the flow. We provide several applications of these results – in complex geometry, optimal transport, and affine geometry.
We study new examples of translating solitons of the mean curvature flow, especially in Minkowski space. We consider for this purpose manifolds admitting submersions and cohomegeneity one actions by isometries on suitable open subsets. This general setting also covers the classical Euclidean examples. As an application, we completely classify timelike, invariant translating solitons by rotations and boosts in Minkowski space.
We continue the study of topological nonrelativistic quantum gravity associated with a family of Ricci flow equations on Riemannian manifolds. This topological gravity is of the cohomological type, and it exhibits an N=2 extended BRST symmetry. In our previous work, we constructed this theory in a two-step procedure in the appropriate nonrelativistic N=2 superspace, first presenting a topological theory of the spatial metric gij, and then adding the superspace versions of the lapse and shift variables n and ni while gauging the symmetries of foliation-preserving spacetime diffeomorphisms. In the relation to Perelman's theory of the Ricci flow, the role of Perelman's dilaton is played by our nonprojectable lapse. Here we demonstrate that this construction is equivalent to a standard one-step BRST gauge-fixing of a theory whose fields are gij, ni and n, and whose gauge symmetries consist of (i) the topological deformations of gij, and (ii) the ultralocal nonrelativistic limit of spacetime diffeomorphisms. The supercharge Q of our superspace construction plays the role of the BRST charge. The spacetime diffeomorphism symmetries appear in an interestingly "shifted" form, which may be of broader interest for nonrelativistic quantum gravity outside of the present topological context. In contrast to the foliation-preserving spacetime diffeomorphisms, the gauge symmetries identified in this paper act nonprojectably on time, making it clear that this theory has no local propagating degrees of freedom. We point out an intriguing dual interpretation of the same theory, as a gauge fixing of a dual copy of ultralocal spacetime diffeomorphisms, with the role of ghosts and antighosts interchanged and the second supercharge Qˉ of the N=2 superalgebra playing the role of the BRST charge in the dual picture.
In this paper, we investigate closed strictly convex hypersurfaces in Rn+1 which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere S+n+1 with power greater than or equal to 1.
We prove the multiplicity one theorem for min-maxfree boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free boundary hypersurface with prescribed mean curvature, which includes the regularity theory for minimizers, compactness theory, and a generic min-max theory with Morse index bounds. As applications, we construct new free boundary minimal hypersurfaces in the unit balls in Euclidean spaces and self-shrinkers of the mean curvature flows with arbitrarily large entropy.
In this paper we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call quotient almost Yamabe solitons because they extend quite naturally those called quotient Yamabe solitons. We then present sufficient conditions for a compact quotient almost Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact quotient gradient almost Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.
We show that the product of two round shrinking spheres is an isolated self-shrinker in any codimension, modulo rotations. Moreover we prove explicit Łojasiewicz inequalities near such products. Łojasiewicz inequalities were previously used by Schulze to prove uniqueness of tangent mean curvature flows at compact shrinkers; our results provide an explicit rate of convergence to products of two spheres.
We establish Łojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, in any codimension. We deduce the uniqueness of blowups at singularities modelled on this class of cylinders, and that any such cylinder is isolated in the space of self-shrinkers. The Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses direct perturbative analysis of the shrinker mean curvature, so it is new even for round cylinders.
We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow on Riemannian manifolds. First we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric, with spatial diffeomorphism invariance and no gauge symmetry, associated with Hamilton's Ricci flow: Hamilton's flow equation appears as the localization equation of the primitive theory. Then we extend the primitive theory by gauging foliation-preserving spacetime symmetries. Crucially, all our theories are required to exhibit an N=2 extended BRST symmetry. First, we gauge spatial diffeomorphisms, and show that this gives us access to the mathematical technique known as the DeTurck trick. Finally, we gauge foliation-preserving time reparametrizations, both with the projectable and nonprojectable lapse function. The path integral of the full theory is localized to the solutions of Ricci-type flow equations, generalizing those of Perelman. The role of Perelman's dilaton is played by the nonprojectable lapse function. Perelman's F-functional appears as the superpotential of our theory. Since there is no spin-statistics theorem in nonrelativistic quantum field theory, the two supercharges of our gravity theory do not have to be interpreted as BRST charges and, after the continuation to real time, the theory can be studied as a candidate for nonrelativistic quantum gravity with propagating bosonic and fermionic degrees of freedom.
Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the (m,ρ)-quasi-Einstein structure on a contact metric manifold. First, we prove that if a K-contact or Sasakian manifold M2n+1 admits a closed (m,ρ)-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature2n(2n+1), and for the particular case – a non-Sasakian (k,μ)-contact structure – it is locally isometric to the product of a Euclidean space \RRn+1 and a sphere Sn of constant curvature 4. Next, we prove that if a compact contact or H-contact metric manifold admits an (m,ρ)-quasi-Einstein structure, whose potential vector field V is collinear to the Reeb vector field, then it is a K-contact η-Einstein manifold.
In this paper, we study the skew mean curvature flow. The results are threefold. First, we prove the global regularity of solutions with initial data which are small perturbations of planes in Sobolev spaces. Second, we prove the modified scattering and the existence of wave operators for small data, which completely determines the set of asymptotic states. Third, we study the Cauchy problem for arbitrary large data.
For an H>0rotationally symmetric embedded torus N0⊂R3, evolved by Inverse Mean Curvature Flow, we show that the total curvature ∣A∣ remains bounded up to the singular time Tmax. We then show convergence of the Nt to a C1 rotationally symmetric embedded torus NTmax as t→Tmax without rescaling. Later, we observe a scale-invariant L2 energy estimate on any embedded solution of the flow in R3 that may be useful in ruling out curvature blowup near singularities in general.
We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimates along the Ricci flow. It has various applications, including to show the continuous dependence of the Ricci flow with respect to the initial metric in Gromov-Hausdorff topology with Ricci curvature bounded below, and to show the compactness of the moduli of Kähler manifolds with bounded scalar curvature and a rough locally almost Euclidean condition.
In this article, we establish a monotonicity formula of Hamilton type entropy along Ricci flow on compact surfaces with boundary. We also study the relation between our entropy functional and the W-functional of Perelman type.
We show that integrability of an almost complex structure in complex dimension m is equivalent, in the presence of an almost hermitian metric, to m(m−1) equations involving what we call shear operators. Inspired by this, we give an ansatz for Kähler metrics in dimension m>1, for which at most m−1 of these shear equations are non-trivial. The equations for gradient Kähler-Ricci solitons in this ansatz are frame dependent PDEs, which specialize to ODEs under extra assumptions. Metrics solving the latter system include a restricted class of cohomogeneity one metrics, and we find among them complete expanding gradient Kähler-Ricci solitons under the action of the (2m−1)-dimensional Heisenberg group, and some incomplete steady solitons. We examine curvature properties and asymptotics for the former Ricci solitons. In another special case of the ansatz we present, for m=2, a class of complete metrics of a more general type which we call gradient Kähler-Ricci skew-solitons, which are cohomogeneity one under the Euclidean plane group action. This paper continues research started in [MR, AM2].
We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n≥4, we find a family of Z2×O(n−1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.
In this paper we prove a Minkowski-like inequality for an asymptotically flat static Einstein-Maxwell (electrostatic) space-time using as approach the inverse mean curvature flow (IMCF). Moreover, we discuss the importance of this inequality in the understanding of the photon sphere.
Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on M which is generated by a Schrödinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.
A recent paper [CGT] studies the evolution of star-shaped mean convex hypersurfaces of the Euclidean space by a class of nonhomogeneous expanding curvature flows. In the present paper we consider the same problem in the real, complex and quaternionic hyperbolic spaces, investigating how the richer geometry of the ambient space affects the evolution. In every case the initial conditions are preserved and the long time existence of the flow is proven. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space, otherwise it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every cases, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.
In this paper, we prove that a Riemannian n-manifold M with sectional curvature bounded above by 1 that contains a minimal 2-sphere of area 4π which has index at least n−2 has constant sectional curvature 1. The proof uses the construction of ancientmean curvature flows that flow out of a minimal submanifold. As a consequence we also prove a rigidity result for the Simon-Smith minimal spheres.
We show that, up to biholomorphism, there is at most one complete Tn-invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold M. We also establish uniqueness without assuming Tn-invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra t of Tn. As an application, we show that, up to isometry, the unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on CP1×C is the standard product metric associated to the Fubini-Study metric on CP1 and the Euclidean metric on C.
Brakke flow is defined with a variational inequality, which means it may have discontinuous mass over time, i.e. have mass drop. It has long been conjectured that the Brakke flow associated to a nonfattening level set flow has no mass drop and achieves equality in the Brakke inequality. Under natural assumptions, we show that a flow has no mass drop if and only if it satisfies the multiplicity one conjecture Hn-a.e. One application is that there is no mass drop for level set flows with mean convex neighborhoods of singularities, and a generic flow has no mass drop until there is a higher multiplicity planar tangent flow. Also, if a nonfattening flow has no higher multiplicity planes as limit flows, then each limit flow has no mass drop. We upgrade these results to equality in the Brakke inequality for certain important cases. We show that nonfattening flows with three-convex blow-up type are Brakke flows with equality. This includes flows with generic singularities in dimension three and flows with mean convex neighborhoods of singularities in dimension four.
In this paper we prove that the Sobolev inequality, the logarithmic Sobolev inequality, the Schrödinger heat kernel upper bound, the Faber-Krahn inequality, the Nash inequality and the Rozenblum-Cwikel-Lieb inequality all equivalently exist on complete gradient shrinking Ricci solitons. We also obtain some integral gap theorems for compact shrinking Ricci solitons.
We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial metric which is close in Lp∩L∞, for any p∈(1,n), where n is the dimension of the manifold. In particular, our result applies to all known examples of 4-dimensional gravitational instantons. Our decay rates are strong enough to prove positive scalar curvature rigidity in Lp, for each p∈[1,n−2n), generalizing a result by Appleton.
In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to 3-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space formS3/Γ or weakly dimension reduce to the 3-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE 4-manifolds or dimension reduce to 3-dimensional manifolds. As an application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE Ricc-flat 4-manifolds.
In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in R3. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the α-Gaussian curvature flow for α∈(0,1/4). We remark that our result does not assume any embeddedness condition.
In this paper, we study the uniformly strong convergence of Kähler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of Kähler-Ricci solitons in sense of diffeomorphism orbits. The result generalizes Tian-Zhu's theorem for the uniqueness of of Kähler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun's result of for the uniqueness of of Kähler-Einstein metric orbits.
We prove that the sharp Li-Yau equality holds for the conjugate heat kernel on shrinking Ricci solitons without any curvature or volume assumptions. This quantity yields several estimates which allows us to classify four dimensional, non-compact shrinking Ricci solitons, which arise as Type I singularity models to the Ricci flow.
We extend some convergence results on nonsingular compact Ricci flows in the papers, and to certain infinite volume noncompact cases which are "partially" nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.
For any flag manifold M=G/K of a compact simple Lie groupG we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on M, and by [BöLS17] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M)=1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system's phase space.
In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic structure on 3-manifolds via ideal triangulation.
Combinatorial Ricci flow on a cusped 3-manifold is an analogue of Chow-Luo's combinatorial Ricci flowon surfaces and Luo's combinatorial Ricci flow on compact 3-manifolds with boundary for finding complete hyperbolic metrics on cusped 3-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped 3-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow through the potential singularities using Luo-Yang's extension. It is shown that the existence of a complete hyperbolic metric on a cusped 3-manifold is equivalent to the convergence of the extended combinatorial Ricci flow, which gives a new characterization of existence of a complete hyperbolic metric on a cusped 3-manifold dual to Casson and Rivin's program. The extended combinatorial Ricci flow also provides an effective algorithm for finding complete hyperbolic metrics on cusped 3-manifolds.
We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flowself-shrinkers and expanders, translating solitons, and rotating solitons. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023. We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is C2 and convex in the gradient variable. We next show that for merely Hölder continuous phases, convex solutions are regular if they are C1,β for sufficiently large β. Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.
We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.
In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension ≥4 in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension ≥4. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected Lp-curvature bounds. As an application of our theory, we obtain a description of the singularity formation of a Ricci flow at its first singular time and a thick-thin decomposition characterizing the long-time behavior of immortal flows. These results generalize Perelman's results in dimension 3 to higher dimensions. We also obtain a Backwards Pseudolocality Theorem and discuss several other applications.
We study almost Riemann solitons and almost Ricci solitons in an (α,β)-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton is pointwise collinear with the structure vector field.
Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analog of Chow-Luo's combinatorial Ricci flow on a triangulated surface and conjectured to find algorithmically the complete hyperbolic metric on the compact 3-manifold with totally geodesic boundary. In this paper, we prove Luo's conjecture affirmatively by extending the combinatorial Ricci flow through the singularities of the flow if the ideally triangulated compact 3-manifold with boundary admits such a metric.
We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian surfaces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n−1)/2 squares. We prove that our discrete FAC is equivalent to the FAC for surfaces with self-reverse metric. If the surface is a disk, the discrete FAC follows from Steinitz's algorithm for transforming curves into pseudolines. This gives a new proof of the FAC for disks with self-reverse metric. We also imitate Ivanov's proof of the same fact, using discrete differential forms. And we prove that the FAC holds for Möbius bands with self-reverse metric. For this we use a combinatorial curve shortening flow developed by de Graaf–Schrijver and Hass–Scott. With the same method we prove the systolic inequality for Klein bottles with self-reverse metric, conjectured by Sabourau–Yassine. Self-reverse metrics can be discretized using walls because every normed plane satisfies Crofton's formula: the length of every segment equals the symplectic measure of the set of lines that it crosses. Directed 2-dimensional metrics have no Crofton formula, but can be discretized as well. Their discretization is a triangulation where the length of each edge is 1 in one way and 0 in the other, and the area of the surface is the number of triangles. This structure is a simplicial set, dual to a plabic graph. The role of the walls is played by Postnikov's strands.
We prove that for any Q-Fano variety X, the special R-test configuration that minimizes the H-functional is unique and has a K-semistableQ-Fano central fibre (W,ξ). Moreover there is a unique K-polystable degeneration of (W,ξ). As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for Kähler-Ricci flow limits on Fano manifolds, which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial Kähler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals of real valuations, which are analogous to the minimization problem for normalized volumes.
We derive a local ansatz for generalized Kähler surfaces with nondegenerate Poisson structure and a biholomorphic S1 action which generalizes the classic Gibbons-Hawking ansatz for invariant hyperKähler manifolds, and allows for the choice of one arbitrary function. By imposing the generalized Kähler-Ricci soliton equation, or equivalently the equations of type IIB string theory, the construction becomes rigid, and we classify all complete solutions with the smallest possible symmetry group.
We prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second fundamental form which we utilize, via a compactness argument, to obtain a convexity estimate. Together, the convexity and cylindrical estimates yield a partial classification of singularity models. We also obtain new rigidity results for ancient solutions.
In this paper we characterize the Einstein metrics in such broader classes of metrics as almost η-Ricci solitons and η-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an η-Ricci soliton is Einstein metric if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost η-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of η-Ricci solitons and gradient η-Ricci solitons, which illustrate our results.
We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
We prove a conjecture of Bernstein that the superconvexity of the heat kernel on hyperbolic space holds in all dimensions and, hence, there is an analog of Huisken's monotonicity formula for mean curvature flow in hyperbolic space of all dimensions.
In this article, we study the Ricci flowneckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion components. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x}×\spheren. We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.
We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.
We develop a compactness theory for super Ricci flows, which lays the foundations for the partial regularity theory in [Bam20b]. Our results imply that any sequence of super Ricci flows of the same dimension that is pointed in an appropriate sense subsequentially converges to a certain type of synthetic flow, called a metric flow. We will study the geometric and analytic properties of this limiting flow, as well as the convergence in detail. We will also see that, under appropriate local curvature bounds, a limit of Ricci flows can be decomposed into a regular and singular part. The regular part can be endowed with a canonical structure of a Ricci flow spacetime and we have smooth convergence on a certain subset of the regular part.
Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants δm whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these δm-invariants characterizes uniform Ding stability. A basic question since Fujita-Odaka's work has been to find an analytic interpretation of these invariants. We show that each δm is the coercivity threshold of a quantized Ding functional on the m-th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for Pn. Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler-Ricci solitons (and the more general g-solitons of Berman-Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.
In this expository note we describe important examples of Lagrangian mean curvature flow in C2 which are invariant under a circle action. Through these examples, we see compact and non-compact situations, long-time existence, singularities forming via explicit models, and significant objects in Riemannian and symplectic geometry, including the Clifford torus, Chekanov torus, Whitney sphere and Lawlor necks.
We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in Rn+1 with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as t→∞. Our approach is based on a localized version of the parabolic maximum principle. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in Rn+1. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in R3.
In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant that only depends on the dimension and possibly a lower scalar curvature bound. In the special case in which the flow consists of Einstein metrics, these bounds agree with the optimal bounds for spaces with Ricci curvature bounded from below. Moreover, our bounds are local in the sense that if a bound depends on the collapsedness of the underlying flow, then we are able to quantify this dependence using the pointed Nash entropy based only at the point in question. Among other things, we will show the following bounds: Upper and lower volume bounds for distance balls, dependence of the pointed Nash entropy on its basepoint in space and time, pointwise upper Gaussian bound on the heat kernel and a bound on its derivative and an L1-Poincaré inequality. The proofs of these bounds will, in part, rely on a monotonicity formula for a notion, called variance of conjugate heat kernels. We will also derive estimates concerning the dependence of the pointed Nash entropy on its basepoint, which are asymptotically optimal. These will allow us to show that points in spacetime that are nearby in a certain sense have comparable pointed Nash entropy. Hence the pointed Nash entropy is a good quantity to measure local collapsedness of a Ricci flow Our results imply a local ε-regularity theorem, improving a result of Hein and Naber. Some of our results also hold for super Ricci flows.
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics' in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kähler-Ricci flow. This leads to global convergence results, and applications to complex geometry. A purely mathematical introduction to the physical idea of T-duality is given, and a discussion of its relationship to generalized Ricci flow.
Ge-Jiang (Geom Funct Anal 27:1231-1256, 2017) proved global ε-regularity for 4-dimensionalRicci flow with bounded scalar curvature. In this note, we extend this result to 4-dimensional Ricci flow with integral bound on the scalar curvature.
We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled Sasakian eta-Einstein metric to the induced metric of the zero section.
We consider almost Riemann solitons (V,λ) in a Riemannian manifold and underline their relation to almost Ricci solitons. When V is of gradient type, using Bochner formula, we explicitly express the function λ by means of the gradient vector field V and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.
In this paper, we study the flow of closed, starshaped hypersurfaces in Rn+1 with speed rασ21/2, where σ21/2 is the normalized square root of the scalar curvature, α≥2, and r is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When α<2, a counterexample is given for the above convergence.
We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric g0 on R4 with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If g0 has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric gTNUT in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when g0 is asymptotically flat. In order to identify infinite-time singularity models we obtain a uniqueness result for gTNUT.
We consider almost Riemann and almost Ricci solitons in a D-homothetically deformed Kenmotsu manifold having as potential vector field a gradient vector field, a solenoidal vector field or the Reeb vector field of the deformed structure, and explicitly obtain the Ricci and scalar curvatures for some cases. We also provide a lower bound for the Ricci curvature of the initial Kenmotsu manifold when the deformed manifold admits a gradient almost Riemann or almost Ricci soliton.
We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.
We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.
We establish a framework, namely, nuclear bounded Fréchet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.
We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems. Our results generalize earlier rigidity results for gradient Ricci solitons and warped productEinstein metrics. In particular, our results apply to homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics.
The target space of the non-linear σ-model is a Riemannian manifold. Although it can be any Riemannian metric, there are certain physically interesting geometries which are worth to study. Here, we numerically evolve the time-symmetric foliations of a family of spherically symmetricasymptotically flat wormholes under the 1-loop renormalization group flow of the non-linear σ-model, the Ricci flow, and under the 2-loop approximation, RG-2 flow. We rely over some theorems adapted from the compact case for studying the evolution of different wormhole types, specially those with high curvature zones. Some metrics expand and others contract at the beginning of the flow, however, all metrics pinch-off at a certain time. This is related with the fact that the flow does not converge to a fixed point when its starting geometry is the spatial sections of a Morris-Thorne wormhole, and therefore the corresponding non-linear σ-model is non-integrable/renormalizable. We present a numerical study of the evolution of wormhole singularities in three dimensions extending the theoretical estimations. Finally, we compute the evolution of the Hamilton's entropy and the Brown-York energy.
In this paper, we extend the theory of Ricci flows satisfying a Type-Iscalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci soliton with singularities of codimension 4. We prove that the entropy of a conjugate heat kernel based at the singular time converges to the soliton entropy of the singular soliton, and use this to characterize the singular set of the Ricci flow solution in terms of a heat kernel density function. This generalizes results previously only known with the stronger assumption of a Type-I curvature bound. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities
This note introduces a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi for submanifolds of Euclidean space. Several properties are proved for this quantity including monotonicity along mean curvature flow in low dimensions and a connection with the conformal volume.
We introduce a new functional inspired by Perelman's λ-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted λALE. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that λALE is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to λALE around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations.
We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.
Let X0,X be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve X0 into X. It is proved that this flow exits globally, preserves both the local convexity and the elastic energy of the evolving curve. If the two curves have same elastic energy then the curvature flow deforms the evolving curve into the target curve X as time tends to infinity.
A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.
In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method which was previously used by Davies in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.
In this paper, we prove that 2 dimensional transversal small perturbations of d-dimensional Euclidean planes under the skew mean curvature flow lead to global solutions which converge to the unperturbed planes in suitable norms. And we clarify the long time behaviors of the solutions in Sobolev spaces.
We prove using an integral criterion the existence and completeness of the wave operators W±(Δh(k),Δg(k),Ig,h(k)) corresponding to the Hodge Laplacians Δν(k) acting on differential k-forms, for ν∈{g,h}, induced by two quasi-isometric Riemannian metrics g and h on a complete open smooth manifold M. In particular, this result provides a criterion for the absolutely continuous spectraσac(Δg(k))=σac(Δh(k)) of Δν(k) to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion requires local curvature bounds and some upper local control on the heat kernel acting on functions provided the Weitzenböck curvature endomorphism is in the Kato class, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.
Surfaces that evolve by mean curvature flow develop singularities. These singularities can be modeled by self-shrinkers, surfaces that shrink by dilations under the flow. Singularities modeled on classical self-shrinkers, namely spheres and cylinders, are stable under perturbations of the flow. In contrast, singularities modeled on other self-shrinkers, such as the Angenent torus, are unstable: perturbing the flow will generally change the kind of singularity. One can measure the degree of instability by computing the Morse index of the self-shrinker, viewed as a critical point of an appropriate functional. In this paper, we present a numerical method for computing the index of rotationally symmetric self-shrinkers. We apply this method to the Angenent torus, the first known nontrivial example of a self-shrinker. We find that, excluding dilations and translations, the index of the Angenent torus is 5, which is consistent with the lower bound of 3 from the work of Liu and the upper bound of 29 from our earlier work. Also, we unexpectedly discover two additional variations of the Angenent torus with eigenvalue−1.
We prove three results in this paper. First, we prove for a wide class of functions φ∈C2(Sn−1) and ψ(X,ν)∈C2(Rn+1×Hn), there exists a unique, entire, strictly convex, spacelike hypersurface Mu satisfying σk(κ[Mu])=ψ(X,ν) and u(x)→∣x∣+φ(∣x∣x) as ∣x∣→∞. Second, when k=n−1,n−2, we show the existence and uniqueness of entire, k-convex, spacelike hypersurface Mu satisfying σk(κ[Mu])=ψ(x,u(x)) and u(x)→∣x∣+φ(∣x∣x) as ∣x∣→∞. Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitonsMu with prescribed asymptotic behavior at infinity for σkcurvature flow equations. Moreover, we prove that the downward translating solitons Mu have bounded principal curvatures.
Assuming that there exists a translating solitonu∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to u∞+Ct as t→∞. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of Ω and Ricci curvature in Ω.
In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space (Rm+p,e−∣x∣2/m\olg) where (Rm+p,\olg) is the standard Euclidean space and x∈Rm+p denotes the position vector. Note that, as a special Riemannian manifold, (Rm+p,e−∣x∣2/m\olg) has an unbounded curvature. Up to a family of diffeomorphisms on Mm, the mean curvature flow we considered here turns out to be equivalent to a special variation of the "{\em conformal mean curvature flow}\," which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to m, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval [0,T) of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance.
We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.
On a compact strictly pseudoconvex CR manifold (M,þ), we consider the CR Yamabe constant of its infinite conformal covering. By using the maximum principles, we then prove a uniqueness theorem for the CR Yamabe flow on a complete noncompact CR manifold. Finally we obtain some properties of the CR Yamabe soliton on complete noncompact CR manifolds.
We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some collapsing and convergence results.
In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles ≤π.
On a smooth n-manifold M with n≥3, we study pairs (g,T) consisting of a Riemannian metric g and a unit length closed vector field T. Motivated by how Ricci solitons generalize Einstein metrics via a distinguished vector field, we propose to generalize space forms by considering those pairs (g,T) whose corresponding Lorentzian metric gL=g−2T♭⊗T♭ has constant curvature. We show by examples that such pairs exist when M is noncompact, and that complete metrics exist among them. When M is compact, however, the situation is more rigid. In the compact setting, we prove that the only pairs (g,T) whose corresponding Lorentzian metric gL is a space form are those where (M,g) is flat and its universal covering splits isometrically as a product R×N. The nonexistence of compact Lorentzian spherical space forms plays a key role in our proof.
Democritus and the early atomists held that "the material cause of all things that exist is the coming together of atoms and void. Atoms are eternal and have many different shapes, and they can cluster together to create things that are perceivable. Differences in shape, arrangement, and position of atoms produce different phenomena". Like the atoms of Democritus, the Grim Reaper solution to curve shortening flow is eternal and indivisible – it does not split off a line, and is itself its only "asymptotic translator". Confirming the heuristic described by Huisken and Sinestrari [J. Differential Geom. 101, 2 (2015), 267-287], we show that it gives rise to a great diversity of convex ancient and translating solutions to mean curvature flow, through the evolution of families of Grim hyperplanes in suitable configurations. We construct, in all dimensions n≥2, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. The latter resolve a conjecture of White [J. Amer. Math. Soc. 16, 1 (2003), 123-138]. We also provide a detailed asymptotic analysis of convex ancient solutions in slab regions in general. Roughly speaking, we show that they decompose "backwards in time" into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. An analogous decomposition holds "forwards in time" for eternal solutions. One consequence is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the mid-plane of their slab.
We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum's result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Finally, we show some applications of the theory to Ricci flows with bounded scalar curvature.
We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only depend on an upper bound for the logarithmic derivative of the volume measure. In particular, our estimates hold for any Ricci flow with scalar curvature bounded below, and such a lower bound of course depends only on the initial data.
We show that flag manifold σ-models (including CPn−1, Grassmannian models as special cases) and their deformed versions may be cast in the form of gauged bosonic Thirring/Gross-Neveu-type systems. Quantum mechanically the gauging is violated by chiral anomalies, which may be cancelled by adding fermions. We conjecture that such models are integrable and check on some examples that the trigonometrically deformed geometries satisfy the generalized Ricci flow equations.
The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur-type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.
On gradient shrinking Ricci solitons, we observe that the study of Schrödinger heat kernel seems to be more natural than the classical heat kernel. In this paper we derive sharp Gaussian upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons. As applications, we prove sharp upper bounds for the Green's function of the Schrödinger operator. We also prove sharp lower bounds for eigenvalues of the Schrödinger operator. These sharp cases are all achieved at Euclidean Gaussian shrinking Ricci solitons.
In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton R4, or S3×R, or S2×R2. In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.
We study mean curvature flow in SKn+1, the round sphere of sectional curvature K>0, under the quadratic curvature pinching condition ∣A∣2<n−21H2+4K when n≥4 and ∣A∣2<53H2+38K when n=3. This condition is related to a famous theorem of Simons, which states that the only minimal hypersurfaces satisfying ∣A∣2<nK are the totally geodesic hyperspheres. It is related to but distinct from two-convexity. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a cylindrical estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari. We conclude that any smoothly, properly, isometrically immersed hypersurface M of SKn+1 satisfying the pinching condition is diffeomorphic to Sn or the connected sum of a finite number of copies of S1×Sn−1. If M is embedded, then it bounds a 1-handlebody. The results are sharp when n≥4.
In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not necessarily gradient solitons while, in the complete non-compact case, we restrict our attention to steady and shrinking gradient solitons. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry a Riemannian manifold equipped with a smooth map φ, called φ-curvatures, which are a natural generalization of the standard curvature tensors in the setting of harmonic-Ricci solitons.
We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel Lp−Lq decay estimates are established, allowing the Schrödinger operator to have a non-trivial L2-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.
In this paper, we study the stability of neckpinch singularities. We show that if a mean curvature flow{Mt} develops only finitely many neckpinch singularities at the first singular time, then the mean curvature flow starting at any sufficiently small perturbation of M0 can also develop only neckpinch type singularities at the first singular time. We also show stability of nondegenerate neckpinch singularities in the above sense, which speaks in favor of stability of Type I singularities.
We consider the evolution by mean curvature of smooth n-dimensional submanifolds in Rn+k which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case k≥2. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.
There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in 2n+ε dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in non-linear sigma models in 2+ε dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of non-linear sigma models in 2+ε dimension from the seminal work by Perelman on the Ricci flow.
A general purely crystallinemean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres.
We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone.
Following the recent observation that the Ricci flow and the infinite distance swampland conjecture are closely related to each other, we will investigate in this paper geometric flow equations for AdS space-time geometries. First, we consider the so called Yamabe and Ricci-Bourguignon flows and we show that these two flows - in contrast to the Ricci flow - lead to infinite distance fixed points for product spaces like AdSd×Sp, where AdSd denotes d-dimensional AdS space and Sp corresponds to a p-dimensional sphere. Second, we consider black hole geometries in AdS space time geometries and their behaviour under the Yamabe and Ricci-Bourguignon flows. Specifically we will examine if and how the AdS black holes will undergo a Hawking-Page phase transition under the Ricci flow, the Yamabe flow and under the general Ricci-Bourguignon flow.
We study the Ricci flow of the four-parameter family of Sp(n+1)-invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. These (non-isometric) ancient solutions all have a larger isometry group, namely Sp(n+1)Sp(1), Sp(n+1)U(1), or U(2n+2). Two ancient solutions are non-collapsed and converge, under the backwards flow, to Jensen's second Einstein metric. One solution parametrizes the well known Berger metrics. The rest are new and collapse, under a rescaling of the backwards flow, to Ziller's second homogeneous Einstein metric on complex projective space.
In this work we establish long-time existence of the normalized Yamabe flow with positive Yamabe constant on a class of manifolds that includes spaces with incomplete cone-edge singularities. We formulate our results axiomatically, so that our results extend to general stratified spaces as well, provided certain parabolic Schauder estimates hold. The central analytic tool is a parabolic Moser iteration, which yields uniform upper and lower bounds on both the solution and the scalar curvature.
Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary kltsingularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler-Ricci/Mabuchi solitons or Kähler-Einstein metrics.
We provide a curvature flow approach to the regular Christoffel-Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term.
We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in R^n+1 is close to a constant in L^n-sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem and using it we are able to show that in R^2 and R^3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by weak solution we mean a flat flow, obtained via the minimizing movements scheme.
In this expository paper, we present a survey about the history of the geometrization conjecture and the background material on the classification of Thurston's eight geometries. We also discuss recent techniques for immersive visualization of relevant three-dimensional manifolds in the context of the Geometrization Conjecture.
Let π:Cn×C→C be the projection map onto the second factor and let D be a domain in Cn+1 such that for y∈π(D), every fiber Dy:=D∩π−1(y) is a smoothly bounded strongly pseudoconvex domain in Cn and is diffeomorphic to each other. By Chau's theorem, the Kähler-Ricci flow has a long time solutionωy(t) on each fiber Dy. This family of flows induces a smooth real (1,1)-form ω(t) on the total space D whose restriction to the fiber Dy satisfies ω(t)∣Dy=ωy(t). In this paper, we prove that ω(t) is positive for all t>0 in D if ω(0) is positive. As a corollary, we also prove that the fiberwise Kähler-Einstein metric is positive semi-definite on D if D is pseudoconvex in Cn+1.
We show that every n-dimensional, κ-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for n=4 or n≥12 has weakly PIC2 and bounded curvature. Combining this with earlier results, we prove that any such solution is isometric to either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton. Also, we classify all complex 2-dimensional, κ-noncollapsed, complete ancient solutions to the Kähler Ricci flow with weakly PIC.
In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.
This paper deals with locally constrainedinverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows we prove long time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases new Minkowski type inequalities are the consequence. In higher dimensions we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.
In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in R3. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in R3 are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in R3 in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in R3. Explicit families of exact solutions for the MCF by cylindrical surfaces in R3 are also given.
The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical approach is particularly suited for such problems for degenerate geometries like the Heisenberg group. In this situation a new type of singularities absent for the Euclidean mean curvature flow occurs, the so-called characteristic points. This paper investigates the asymptotic behaviour of the regularised optimal controls in the vicinity of such characteristic points.
We point out that Ecker's local integral quantity agrees with Huisken's global integral quantity at infinity for ancientmean curvature flows if Huisken's one is finite on each time-slice. In particular, this means that the finiteness of Ecker's integral quantity at infinity implies the finiteness of the entropy at infinity.
A recent article by Li and Lv considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.
In Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation H=α⟨N,x⟩+λ, where N is the Gauss map, x is the position vector and α and λ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.
Let C be an m-dimensional cone immersed in Rn+m. In this paper, we show that if F:Mm→Rn+m is a properly immersed mean curvature flowself-shrinker which is smoothly asymptotic to C, then it is unique and converges to C with unit multiplicity. Furthermore, if F1 and F2 are self-expanders that both converge to C smoothly asymptotically and their separation decreases faster than ρ−m−1e−ρ2/4 in the Hausdorff metric, then the images of F1 and F2 coincide.
We extend the second part of on the uniqueness of ancientκ-solutions to higher dimensions. In dimensions n≥4, an ancient κ-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is κ-noncollapsed. We show that the only noncompact ancient κ-solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.
We study self-expanding solutions Mm⊂Rn of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function ∣A∣2/∣H∣2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the pricipal normal ξ=H/∣H∣ is parallel in the normal bundle, then a similar result holds in higher codimension for the function ∣Aξ∣2/∣H∣2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.
The aim of this article is to provide a Liouville theorem for heat equation along ancientsuper Ricci flow. We formulate such a Liouville theorem under a growth condition concerning Perelman's reduced distance.
In this note, we establish a local maximum principle along Ricci flow under scaling invariant curvature condition. This unifies the known preservation of nonnegativity results along Ricci flow with unbounded curvature. By combining with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard's localized version of a maximum principle given by R. Bamler, E. Cabezas-Rivas, and B. Wilking on the lower bound of curvature conditions.
We introduce natural deformation classes of generalized Kähler structures using the Courant symmetry group. We show that these yield natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. Lastly we show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.
In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean n-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a byproduct, two existence results of solutions to the even Orlicz-Minkowski problem are obtained, one of which improves the known result.
It is shown that the Pohlmeyer map of a σ-model with a toric two-dimensional target space naturally leads to the `sausage' metric. We then elaborate the trigonometric deformation of the CPn−1-model, proving that its T-dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold σ-models and Toda field theories.
In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. The mean curvature flow itself is used in the argument presented.
The aim of this paper is characterize a class of contact metric manifolds admitting ∗-conformal Ricci soliton. It is shown that if a (2n+1)-dimensional N(k)-contact metric manifold M admits ∗-conformal Ricci soliton or ∗-conformal gradient Ricci soliton, then the manifold M is ∗-Ricci at and locally isometric to the Riemannian of a flat (n+1)-dimensional manifold and an n-dimensional manifold of constant curvature 4 for n>1 and flat for n=1. Further, for the first case, the soliton vector field is conformal and for the ∗-gradient case, the potential function f is either harmonic or satisfy a Poisson equation. Finally, an example is presented to support the results.
We study the long-time existence and behavior for a class of anisotropic non-homogeneousGauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding Lp version is the even Lp-Minkowski problem for p>−n−1. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the Lp versions are the even Lp-Minkowski problem for p>0 and the Lp-Minkowski problem for p>1. In the final section, we use a curvature flow with no global term to solve a class of Lp-Christoffel-Minkowski type problems.
It is shown that bounds of all orders of derivative would follow from uniform bounds for the metric and the torsion 1-form, for a flow in non-Kähler geometry which can be interpreted as either a flow for the Type IIB string or the Anomaly flow with source term and zero slope parameter. A key ingredient in the proof is a formulation of this flow unifying it with the Ricci flow, which was recently found.
In this article, we provide some volume growth estimates for complete noncompact gradient Ricci solitons and quasi-Einstein manifolds similar to the classical results by Bishop, Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. We prove a sharp volume growth estimate for complete noncompact gradient shrinking Ricci soliton. Moreover, we provide upper bound volume growth estimates for complete noncompact quasi-Einstein manifolds with λ=0. In addition, we prove that geodesic balls of complete noncompact quasi-Einstein manifolds with λ<0 and μ≤0 have at most exponential volume growth.
In this paper, we investigate the mean curvature flow of compact surfaces in 4-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which generalise Baker-Nguyen's convergence theorem.
In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for submanifolds with positive Ricci curvature. Finally, we present a list of unsolved problems in this area.
In this paper, we prove convergence of the high codimension mean curvature flow in the sphere to either a round point or a totally geodesic sphere assuming a pinching condition between the norm squared of the second fundamental form and the norm squared of the mean curvature and the background curvature of the sphere. We show that this pinching is sharp for dimension n≥4 but is not sharp for dimension n=2,3. For dimension n=2 and codimension 2, we consider an alternative pinching condition which includes the normal curvature of the normal bundle. Finally, we sharpen the Chern-do Carmo-Kobayashi curvature condition for surfaces in the four sphere - this curvature condition is sharp for minimal surfaces and we conjecture it to be sharp for curvature flows in the sphere.
We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compacttranslating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time −∞ from two opposite ends.
In this research article, we establish the geometrical bearing on Riemannian submersions in terms of η-Ricci-Yamabe Soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an η-Ricci-Yamabe soliton, η-Ricci soliton and η-Yamabe soliton. We also discuss the various conditions for which the target manifold of Riemannian submersion is an η-Ricci-Yamabe soliton, η-Ricci soliton, η-Yamabe soliton and quasi-Yamabe soliton. In a particular case when the potential filed V of the η-Ricci-Yamabe soliton is of gradient type, we derive a Laplacian equation and providing some examples of an η-Ricci-Yamabe soliton on a Riemannian submersion. Finally, we study harmonic aspect of η-Ricci-Yamabe soliton on Riemannian submersions and mention geometrical and physical effects of Ricci-Yamabe solitons.
The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem, in the context of collapsing Riemannian manifolds with Ricci curvature bounded below: if a closed manifold with Ricci curvature uniformly bounded below is Gromov-Hausdorff close to a (lower dimensional) manifold with bounded geometry, and has the difference of their first Betti numbers equal to the dimensional difference, then it is diffeomorphic to a torus bundle over the one with bounded geometry. We rely on two novel technical tools: the first is an effective control of the spreading of minimal geodesics with initial data parallel transported along a short geodesic segment, and the second is a Ricci flow smoothing result for certain collapsing initial data with Ricci curvature bounded below.
In this article we study Chen's flow of curves from theoreical and numerical perspectives. We investigate two settings: that of closed immersed ω-circles, and immersed lines satisfying a cocompactness condition. In each of the settings our goal is to find geometric conditions that allow us to understand the global behaviour of the flow: for the cocompact case, the condition is straightforward and the argument is largely standard. For the closed case however, the argument is quite complex. The flow shrinks every initial curve to a point if it does not become singular beforehand, and we must identify a condition to ensure this behaviour as well as identify the point in order to perform the requisite rescaling. We are able to successfully conduct a full analysis of the rescaling under a curvature condition. The analysis resembles the case of the mean curvature flow more than other fourth-order curvature flow such as the elastic flow or the curve diffusion flow, despite the lack of maximum and comparison principles. Our work is informed by a numerical study of the flow, and we include a section that explains the algorithms used and gives some further simulations.
We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by F−p with positive power p for a smooth, symmetric, strictly increasing and 1-homogeneous curvature function f of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for 0<p≤1 we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.
In dimensions n≥5, we prove a canonical neighborhood theorem for the mean curvature flow of compact n-dimensional submanifolds in RN satisfying a pinching condition ∣A∣2<c∣H∣2 for c=min{2n(n+2)3(n+1),n−21}.
In this paper, we study the existence of harmonic and bi-harmonic maps into Riemannian manifolds admitting a conformal vector field, or a nontrivial Ricci solitons.
We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the generic perturbation is outward, then the next mean curvature flowsingularity is cylindrical and collapsing from outside; if the generic perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.
We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent.
We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions n≥8. Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to Sn or to a finite connected sum of Sn−1-bundles over S1.
We introduce and study μK-stability of polarized schemes with respect to general test configurations as an algebro-geometric aspect of the existence of μ-cscK metrics, which is introduced in the paper arXiv:1902.00664 as a framework unifying the frameworks of Kähler-Ricci solitons and cscK metrics. This article consists of two ingredients. On one hand, we develop a foundational framework concerning `derivative of relative equivariant intersection', which we call equivariant calculus. A core claim in equivariant calculus is a convergence result for some infinite series in equivariant cohomology, which is obtained by relative equivariant intersections. Our proof is based on some observations on deRham-Cartan model of equivariant locally finite homology. This framework furnishes a language to describe μK-stability. We in particular conclude the μK-semistability of μ-cscK manifolds with respect to general test configurations. On the other hand, we introduce an equivariant character μˇλ called μ-character for equivariant family of polarized schemes, motivated by the μ-volume functional introduced in the paper arXiv1902.00664 as a generalization of Tian-Zhu's functional in the theory of Kähler-Ricci soliton. The equivariant derivative of the μ-character derive μ-Futaki invariant for general test configuration, and furthermore, it also produces an analogue of the equivariant first Chern class of CM line bundle for family of polarized schemes, which is irrational and hence cannot be realized as a Q-line bundle in our general μK-stability setup.
A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms of their entropy and their maximum and minimum radii. While there have been a few lower bound results in the literature, we believe that this result is the first upper bound on the index of a self-shrinker. Our methods also give lower bounds on the index and the entropy, and our methods give simple formulas for two entropy-decreasing variations whose existence was proved by Liu. Surprisingly, the eigenvalue corresponding to these variations is exactly −1. Finally, we present some preliminary results in higher dimensions and six potential directions for future work.
We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d complete Riemannian manifold with non-negative Ricci curvature, there exists a smooth Ricci flow starting from it.
We study gradient Ricci solitonswarped products whose base is the Euclidean space. We show that the warping functions of these manifolds are invariant under the (n-1)-dimensional translation group. We characterize the potential function when the torsion function depends only on one variable, which is a particular case of invariance by translation. From this study, we derive complete examples of expanding gradient Ricci solitons.
We deal with solitons of the mean curvature flow. The definition of translating solitons on a light-like direction in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, translation surfaces, obtaining space-like and time-like, entire and not entire, complete and incomplete examples. Among them, all our time-like examples are incomplete. The second family consists of those which are invariant by a 1-dimensional subgroup of parabolic motions, i.e, with light-like axis. The classification result implies that all examples of this second family have singularities.
In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.
We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, assuming Ilmanen's multiplicity one conjecture and no cylinder conjecture, we conclude that in the two-dimensional case, the diameter always stays bounded. We also obtain sharp Ln−1 bound for the curvature. The key ingredients for our proof are the Lojasiewicz inequalities by Colding-Minicozzi and Chodosh-Schulze, and the solution of the mean-convex neighbourhood conjecture by Choi, Haslhofer, Hershkovits and White. Our results improve the prior results by Gianniotis-Haslhofer, where diameter and curvature control has been obtained under the more restrictive assumption that the flow is globally two-convex.
In this paper we have introduced the notion of ∗−Ricci flow and shown that ∗−Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the ∗− Ricci flow. We have also find the deformation of geometric curvature tensors under ∗− Ricci flow. In the last two section of the paper, we have found the ℑ-functional and ω− functional for ∗− Ricci flow respectively.
A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons in (arXiv:math/0211159). A similar quantity was found by Feldman, Ilmanen, and Ni in 2005 which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.
We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.
We construct eternal mean curvature flows of tori in perturbations of the standard unit sphere S3. This has applications to the study of the Morse homologies of area functionals over the space of embedded tori in S3.
We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.
We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate Lp-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the Lp-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli-Guan-Ma (Commun. Pure Appl. Math. 60 (2007), 1769–1791). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.
We show that the mean curvature flow of generic closed surfaces in R3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.
We extend the arguments of Tosatti-Zhang to reduce a well-known conjecture concerning the structure of the Gromov-Hausdorff limit in both the setting of degenerating Calabi-Yau manifolds and the Kähler-Ricci flow to a certain partial second-order estimate.
We develop a min-max theory for asymptotically conicalself-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there exists a new asymptotically conical self-expander trapped between the two.
We study the following question: What is the largest deterministic amount of time T∗ that a suitably normalized martingale X can be kept inside a convex body K in Rd? We show, in a viscosity framework, that T∗ equals the time it takes for the relative boundary of K to reach X(0) as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio–Soner codimension-(d−1)mean curvature flow of the 1-skeleton of K. Our results are obtained by a mix of probabilistic and analytic methods.
We obtain the instability of Type (II) Lawson-Osserman cones in Euclidean spaces, and thus provide a family of (uncountably many) unstable solutions with singularity to the Dirichlet problem for minimal graphs of high codimension versus smooth unstable ones by Lawson-Osserman through a min-max technique. To our knowledge, these are the first examples of non-smooth unstable minimal graphs and unlikely detectible through the mean curvature flow or min-max theory.
The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean curvature flow, which is the parabolic analogue of the improved convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level sets are converging in C2, then they converge in C2,θ. As an application, we obtain a curvature estimate for parabolic Allen-Cahn equation, which can be viewed as a diffused version of Brakke's and White's regularity theorem for mean curvature flow
Given a noncompact spin manifold M with a fixed topological spin structure and two complete Riemannian metrics g and h on M with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave operators W±(Dh,Dg,Ig,h) and W±(Dh2,Dg2,Ig,h), where Ig,h is the canonically given unitary map between the underlying L2-spaces of spinors. This criterion does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow.
We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a "round point" in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.
In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean \mathbbR^n+1 with speed u^αf^β(α, β\in\mathbbR^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If α\leq 0<β\leq 1-α, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.
The Ricci flow was introduced by Hamilton and gained its importance through the years. Of special importance is the limiting behavior of the flow and its symmetry properties. Taking this into account, we present a novel normalization for the homogeneous Ricci flow with natural compactness properties. In addition, we present a characterization for Gromov-Hausdorff limits of homogeneous spaces. As a result, we present a detailed picture of the homogeneous Ricci flow for three-isotropy-summands flag manifolds: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. Moreover, we achieve a full classification of the possible Gromov-Hausdorff limits of the aforementioned lines of flow.
In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in RN in every dimension N≥8. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension N≥8, a nontrivial subset of these solutions has uniformly bounded mean curvature.
Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of these flows for plane curves with the rotation number one. In particular, when the initial curve is strictly convex, the flow exists globally in time, and converges to a circle as time tends to infinity. Even if the initial curve is not strictly convex, a global solution, if it exists, converges to a circle. Here, we deal with curves with a general rotation number, and show, not only a similar result for global solutions, but also a blow-up criterion, upper estimates of the blow-up time, and blow-up rate from below. For this purpose, we use a geometric quantity which has never been considered before.
Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global n+11-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 21 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.
We prove that in the absence of topological changes, the notion of BV solutions to planar multiphasemean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees that one has reached a global minimum in the energy landscape, the existence of a "gradient flow calibration" ensures that the route of steepest descent in the energy landscape is unique and stable.
In this paper we have investigated some aspects of gradient ρ-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient ρ-Einstein soliton is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have showed that in a non-compact gradient ρ-Einstein soliton satisfying some integral condition, the scalar curvature vanishes.
These notes arose from a mini lecture series the author gave at the Early Career Researchers Workshop on Geometric Analysis and PDEs, held in January 2020 at the Matrix institute of the University of Melbourne. We discussed some classical aspects of expanding curvature flows and obtained first applications. In these notes we will give a detailed account on what was covered during the lectures.
We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of s-fractional mean curvature flows as s→0+ and s→1−. In analogy with the s-fractional mean curvature flows, we introduce the notion of s-Riesz curvature flows and characterize its limit as s→0−. Eventually, we discuss the limit behavior as r→0+ of the flow generated by a regularization of the r-Minkowski content.
Let (g,X) be a Kähler-Ricci soliton on a complex manifold M. We prove that if the Kähler manifold (M,g) can be Kähler immersed into a definite or indefinite complex space form of constant holomorphic sectional curvature 2c, then g is Einstein. Moreover, its Einstein constant is a rational multiple of c.
In this paper, we study the classification of κ-noncollapsed ancient solutions to three-dimensionalRicci flow on S3. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.
We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds V2Rn and V1+k2Rn, with n=1+k2+k3. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
In this paper, we first study the locally constrainedcurvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the mth quermassintegral and decreases (m+1)th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of kth Gauss-Bonnet curvature of a smooth h-convex hypersurface to its mth quermassintegral (for 0≤m≤2k+1≤n), and comparing the weighted integral of kth mean curvature to its mth quermassintegral (for 0≤m≤k≤n). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of kth shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality.
The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.
Existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on R^n.
We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger- Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.
In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.
In this paper, we prove the Hamilton-Tian conjecture for Kähler-Ricci flow based on a recent work of Liu-Székelyhidi on Tian's partical C0-estimate for poralized Kähler metrics with Ricci bounded below. The Yau-Tian-Donaldson conjecture for the existence of Kähler-Einstein metrics on Fano manifolds will be also discussed.
We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated (1,1)-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins-Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a C-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.
The main purpose of this paper is to show that a normalized non-steady gradient Ricci soliton (M,g,f,λ) of dimension n is trivial if and only if its scalar curvature S satisfies the equality S=λ(f+(n/2)).
We consider an anisotropic curvature flow V=A(n)H+B(n) in a band domain Ω:=[−1,1]×R, where n, V and H denote the unit normal vector, normal velocity and curvature, respectively, of a graphic curve Γt. We consider the case when A>0>B and the curve Γt contacts ∂±Ω with slopes equaling to ±1 times of its height (which are unbounded when the solution moves to infinity). First, we present the global well-posedness and then, under some symmetric assumptions on A and B, we show the uniform interior gradient estimates for the solution. Based on these estimates, we prove that Γt converges as t→∞ in Cloc2,1((−1,1)×R) topology to a cup-like traveling wave with {\it infinite} derivatives on the boundaries.
In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation (N2t⋅∂t2∂2+NtR(t,x)∂t∂−Δg~(N)(t))u=R(t,x) outside its wavefront set onto Rt×Mx,g(t) solves the backward heat equation ∂tu+Δx,g(t)u=−R(t,x) within some appropriate time interval. We discuss this approximation starting from Euclidean case, and then extend to the open Riemannian manifold situation. This idea partially comes from Perelman's original papers in proving Poincaré conjecture as well as Terence Tao's Notes in UCLA.
In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to S4 or RP4 or CP2. We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.
We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.
We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics. In order to prove the first theorem, we employ the generalization of Grayson's curve shortening flow developed by Angenent-Oaks.
In this paper, we compute canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensionalLorentzianLie groups with some product structure. We define algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. We classify algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.
In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsingCalabi-Yau metrics, that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges locally smoothly away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.
We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive and the potential function is proper. We then provide a sufficient condition for the scalar curvature of expanding soliton being nonnegative. We also estimate the curvature of expanding soliton in dimensions three and four. As an application, we prove a gap theorem on three dimensional gradient expander.
In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or CD(0,m)-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savaré and Toscani on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein manifolds and a special (K,m)-Ricci flow with Hessian solitons. Inspired by Lu-Ni-Vazquez-Villani, we prove the Aronson-Benilan estimates for NLDE on compact Riemannian manifolds with CD(0,m)-condition. We also prove the NIW formula which indicates an intrinsic relationship between the second order derivative of the Renyi entropy power Np, the p-th Fisher information Ip and the time derivative of the W-entropy associated with NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete Riemannian manifolds with non-negative Ricci curvature or CD(0,m)-condition and maximal volume growth condition.
In this paper, we consider the V-soliton equation which is a degenerate fully nonlinear equation introduced by La Nave and Tian in their work on Kähler-Ricci flow on symplectic quotients. One can apply the interpretation to study finite time singularities of the Kähler-Ricci flow. As in the case of Kähler-Einstein metrics, we can also reduce the V-soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampère type. We formulate some preliminary estimates for such a scalar equation on a compact Kähler manifold M.
We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the boundedness of a space-time integral norm of the scalar curvature.
In this paper, we can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mn×R, where Mn is an n-dimensional (n≥2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.
We use the momentum construction for S1-invariant Kähler metrics as developed by Hwang-Singer to construct new examples of steady Kähler-Ricci solitons. We also prove that these solitons are unique in their Kähler class, provided the vector field and the asymptotic behaviour are fixed.
Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a compactness theorem for gradient shrinking Ricci soliton satisfying some scalar curvature condition. Finally, we have showed that a gradient expanding Ricci soliton with linear volume growth and positive potential function is an Einstein manifold.
Let (X,ω) be a compact Kähler manifold of complex dimension n and (L,h) be a holomorphic line bundle over X. The line bundle mean curvature flow was introduced in in order to find deformed Hermitian-Yang-Mills metrics on L. In this paper, we consider the stability of the line bundle mean curvature flow. Suppose there exists a deformed Hermitian Yang-Mills metric h^ on L. We prove that the line bundle mean curvature flow converges to h^ exponentially in C∞ sense as long as the initial metric is close to h^ in C2-norm.
In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument provides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf.) in Rn×R, n≥2, of the form (r,y(r)) or (r(y),y) where r=∣x∣, x∈Rn, is the radially symmetric coordinate and y∈R. More precisely for any n1<λ<n−11 and μ<0, we will give a new proof of the existence of a unique solution r(y)∈C2(μ,∞)∩C([μ,∞)) of the equation 1+ry(y)2ryy(y)=r(y)n−1−λ(r(y)−yry(y))1+ry(y)2, r(y)>0, in (μ,∞) which satisfies r(μ)=0 and ry(μ)=limy↘μry(y)=+∞. We also prove that there exist constants y2>y1>0 such that ry(y)>0 for any μ<y<y1, ry(y1)=0, ry(y)<0 for any y>y1, ryy(y)<0 for any μ<y<y2, ryy(y2)=0 and ryy(y)>0 for any y>y2. Moreover limy→+∞r(y)=0 and limy→+∞yry(y)=0.
We consider the parabolic polyharmonic diffusion and L2-gradient flows of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L2, then the evolving curve converges exponentially in the C∞ topology to a straight horizontal line segment. The same behaviour is shown for the L2-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.
We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models.
In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if Σ is a closed minimal surface embedded in a Riemannian three-manifold(N,g) and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on Σ, a minimal surface M˘ resembling two copies of Σ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution φ on Σ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area ∣M˘∣ of M˘ in terms of the sizes of its catenoidal bridges and confirm that it is negative; ∣M˘∣<2∣Σ∣ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general (O(2)×Z2)-symmetric backgrounds (Σ,N,g). Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for Σ=S2⊂N=S3 providing new constructions even in that case. In Part III, applying the results of Parts I and II – appropriately modified for the catenoid and the critical catenoid – we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.
We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.
In this paper, we study the precise asymptotics of noncompact Type-IIb solutions to the mean curvature flow. Precisely, for each real number γ>0, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics as t↗∞: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the Type-IIb rate (2t+1)(γ−1)/2. (2) In a neighbourhood of the tip, the Type-IIb blow-up of the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface has a precise growth rate depending on γ.
We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.
We consider the mean curvature flow of compact convex surfaces in Euclidean 3-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds.
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear parabolic equations on compact Riemannian manifolds under the Ricci flow.
In this study, we provide some classifications for half-conformally flat gradient f-almost Ricci solitons, denoted by (M,g,f), in both Lorentzian and neutral signature. First, we prove that if ∣∣∇f∣∣ is a non-zero constant, then (M,g,f) is locally isometric to a warped product of the form I×φN, where I⊂R and N is of constant sectional curvature. On the other hand, if ∣∣∇f∣∣=0, then it is locally a Walker manifold. Then, we construct an example of 4-dimensional steady gradient f-almost Ricci solitons in neutral signature. At the end, we give more physical applications of gradient Ricci solitons endowed with the standard static spacetime metric.
In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the rigidity models of the Shannon entropy power are Einstein or quasi Einstein manifolds with Hessian solitons. Moreover, we prove the convexity of the Shannon entropy power for the conjugate heat equation introduced by G. Perelman on Ricci flow and that the corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with non-negative (Bakry-Emery) Ricci curvature and the maximal volume growth condition.
We show that the isoperimetric profile hg(t)(ξ) of a compact Riemannian manifold (M,g) is jointly continuous when metrics g(t) vary continuously. We also show that, when M is a compact surface and g(t) evolves under normalized Ricci flow, hg(t)2(ξ) is uniform Lipschitz continuous and hence hg(t)(ξ) is uniform locally Lipschitz continuous.