193 papers from 2016, 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.
ξ-submanifold in the Euclidean space \bbrm+p is a natural extension of the concept of self-shrinker to the mean curvature flow in \bbrm+p. It is also a generalization of the λ-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for ξ-submanifolds are established. First, it is shown that a submanifold in \bbrm+p is a ξ-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space (\bbrm+p,e−\fr∣x∣2m\lagl⋅,⋅\ragl); Then, two weighted volume functionals Vξ and Vˉξ are introduced and it is proved that ξ-submanifolds can be characterized as the critical points of these two functionals; Also, the corresponding second variation formulas are computed and the (W-)stability properties for ξ-submanifolds are systematically studied. In particular, it is proved that m-planes are the only properly immersed, complete W-stable ξ-submanifolds with flat normal bundle under a technical condition. It would be interesting if this additional restriction could be removed.
We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a Kähler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.
In this note, we prove that a 3-dimensional steady Ricci soliton is rotationally symmetric if its scalar curvatureR(x) satisfies ρ(x)C0−1≤R(x)≤ρ(x)C0 for some constant C0>0, where ρ(x) denotes the distance from a fixed point x0. Our result doesn't assume that the soliton is κ-noncollapsed.
We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give a curvature estimate for translators in R3 with small entropy. Finally, we estimate the spectrum of the stability operator L for translators and give a rigidity result of L-stable translators.
We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is ispired by [10] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour of a suitable ratio associated to the hypersurface that converges exponentially in time to the value associated to a geodesic sphere.
We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the minimizing movements method we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results.
In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in Rn+1. We study the basic properties, such as positivity preserving property, of the translating mean curvature flow. The Dirichlet problem for the graphical translating mean curvature flow is studied and the global existence of the flow and the convergence property are also considered.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m≥3. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.
This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on Kähler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Ampère equations when the background Kähler metrics on Cn have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on Cn with background metric gβ=−1(dz1∧dz1ˉ+…+β2∣zn∣−2(1−β)dzn∧dznˉ) for β∈(0,1). Our results give an effective elliptic Schauder estimate of Donaldson and a direct proof for the short time existence of the conical Kähler-Ricci flow.
We construct examples of spherical space forms(S3/Γ,g) with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at (S3/Γ,g): a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension 3 a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of singularities which can occur for metrics with these symmetries.
We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the dilaton shift in string theory and prove that the Hull-Strominger system is preserved by T-duality.
In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of σkα-flow must be a round sphere. We also obtain a similar result for the solutions of F=−⟨X,en+1⟩(∗) with a non-homogeneous function F. At last, we prove that if F can be compared with kσk(n−k+1)σk−1, then a closed strictly k-convex solution of (∗) must be a round sphere.
This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature (H>0). Previously Hitchin proved that any compact Kähler surface with H>0 must be rational and he constructed such examples on Hirzebruch surfaces M2,k=P(Hk⊕1CP1). We generalize Hitchin's construction and prove that any Hirzebruch manifold Mn,k=P(Hk⊕1CPn−1) admits a Kähler metric of H>0 in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kähler-Ricci flow on Hirzebruch manifolds are also discussed. It seems interesting to study the space of all Kähler metrics of H>0 on a given Kähler manifold. We give higher dimensional examples such that some Kähler classes admit Kähler metrics with H>0 and some do not.
The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensionalRicci flow, and leads to local C/t decay of the full curvature tensor, irrespective of what is happening beyond the local region. As a by-product, our results generalise the Pseudolocality theorem of Perelman and Tian-Wang in this dimension by not requiring the Ricci curvature to be almost-positive, and not asking the volume growth to be almost-Euclidean.
We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere. Our result is sharp in many interesting situations. For example, the curvature tensors of CPn/2, HPn/4, Sn−k×Sk (2≤k≤n−2), Sn−2×H2, Sn−2×R2 all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of Sn−1×R lies in the interior of our curvature cone.
Given a smooth, symmetric, homogeneous of degree one function f(λ1,⋯,λn) satisfying ∂if>0 for all i=1,⋯,n, and a rotationally symmetric cone C in Rn+1, we show that there is a fself-shrinker (i.e. a hypersurface Σ in Rn+1 which satisfies f(κ1,⋯,κn)+21X⋅N=0, where X is the position vector, N is the unit normal vector, and κ1,⋯,κn are principal curvatures of Σ) that is asymptotic to C at infinity.
In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our main result yields the equivalence of (i) dynamic convexity of the Boltzmann entropy on the (time-dependent) L2-Wasserstein space; (ii) monotonicity of L2-Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time); (iii) gradient estimates for the heat flow acting on functions (forward in time); (iv) a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI-flow for the (time-dependent) energy in L2-Hilbert space and the dual heat flow on probability measures as the unique backward EVI-flow for the (time-dependent) Boltzmann entropy in L2-Wasserstein space.
We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.
In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic 3-space.
Let (M,g0) be a compact n-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over (Sn−1,g). We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. In the case that the initial manifold has isolated singularities asymptotic to a non-negatively curved cone over (Sn−1/Γ,g), where Γ acts freely and properly discontinuously, we extend the above result by showing that starting from such an initial condition there exists a smooth Ricci flow with isolated orbifold singularities.
We prove that all entire smooth strictly convex self-shrinking solutions on Rn to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire self-shrinking solutions on Cn to the Kähler-Ricci flow under certain conditions.
We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson-Lie T-duality is compatible with the 1-loop renormalization group.
This paper concerns the evolution of a closed hypersurface of dimension n(≥2) in the Euclidean space Rn+1 under a mixed volume preserving flow. The speed equals a power β(≥1) of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy and Cabezas-Rivas and Sinestrari to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.
This paper concerns the evolution of a closed convex hypersurface in Rn+1, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.
We generalize the ancient solutions of the Ricci flow on certain principal SO(3) bundles over compact quaternionic Kähler manifolds constructed by Bakas, Kong, and Ni to certain RP3 fibre bundles over a product of two compact quaternionic Kähler manifolds. The ancient solutions are of Type I, κ-noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional non-Kählerian bundle spaces are also constructed.
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.
We derive local C2 estimates for complete non-compacttranslating solitons of the Gauss curvature flow in R3 which are graphs over a convex domain Ω. This is closely is related to deriving local C1,1 estimates for the degenerate Monge-Ampére equation. As a result, given a weakly convex bounded domain Ω, we establish the existence of a Cloc1,1 translating soliton. In particular, when the boundary ∂Ω has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.
Previously the DeTurck 'trick' has been used to render the stationary Einstein's equation a well posed elliptic system that may be solved numerically by geometric flow or directly. Whilst in the static case for pure gravity with zero or negative cosmological constant there is a simple proof that solving the modified "harmonic" Einstein's equation leads to a solution of the original Einstein system - i.e. not a Ricci soliton - in the stationary case this argument no longer works. Here we provide a new argument that extends the static result to the case of stationary spacetimes that possess a "t-φ" reflection symmetry. Defining a "soliton charge" from the asymptotic behaviour of the solution, we show that this quantity is always non-positive. Provided asymptotic conditions are chosen such that this charge vanishes, then stationary solitons cannot exist.
Here we provide a uniqueness result for viscosity solutions to sub-Riemannian mean curvature flow. In this setting the uniqueness cannot be deduced via comparison principle, which is known only for graphs and for radially symmetric surfaces, due to the presence of characteristic points. Here we prove that any viscosity solution is limit of a sequence of solutions of Riemannian flow, and obtain as a consequence uniqueness and comparison principle already known in the approximating riemannian setting. The results are provided in the settings of both 3-dimensional rototranslation group SE(2) and Carnot groups of step 2, which are particularly important due to their relation to surface completion problem of model of the visual cortex.
Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumptions on the warp potential.
We consider the warped product manifold, R+×IdMn, with Riemannian metric γ≡dr2⊕r2σ, where (Mn,σ) is a smooth closed Riemannian n-manifold. We investigate what sufficient curvature condition is required of σ to ensure that a solution to the inverse mean curvature flow - commencing with a star-shaped surface - exists for all timest>0.
We consider the non-trivial Ricci soliton on CP2#CP2 constructed by Koiso and Cao. It is a Kähler metric invariant by the U(2) action on CP2#CP2. We study its Yamabe equation and prove it has exactly one U(2)−invariant solution up to homothecies.
In this article we investigate a family of nonlinear evolutions of polygons in the plane called the β-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable in the sense that nearby polygons shrink to points that rescale to a regular polygon. In dimension four we show that the shape of a square is locally stable under perturbations along a hypersurface of all possible perturbations. Furthermore, we are able to show that under a lower bound on angles there exists a rescaled sequence extracted from the evolution that converges to a limiting polygon that is a self-similar solution of the flow. The last result uses a monotonicity formula analogous to Huisken's for the curve shortening flow.
We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is Pm blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem.
Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.
We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kahler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth background data. We discuss the possibility of extending these results to more singular settings, pointing out a key error in the existing literature on viscosity solutions to complex Monge-Ampere equations/Kahler-Ricci flow.
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.
We prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly, Ricci curvature bounded below by a negative quadratic function, and with almost Euclidean isoperimetric inequality holds locally. In particular, this result applies to manifolds with both Ricci curvature and injectivity radius bounded from below. We also study the short-time behaviour of these solutions which may have unbounded curvature at the initial time, and provide some applications. A key tool is Perelman's pseudolocality theorem.
In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in Hn+1 and show long time existence of the flow as well as asymptotic convergence to horospheres.
We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.
We prove that there does not exist non-constant positive f-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the Lp(p≥1or0<p≤1) Liouville theorems on the complete gradient shrinking Ricci solitons.
In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.
We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor φ(r). If φ′(r)>0 and φ′′(r)≥0, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of φ′′(r) and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for f-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.
In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in Rn+1(n=2,3) is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton Σ with nonnegative scalar curvature, we prove that if the mean curvature of Σ does not change signs on each end, then Σ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.
Let N be a smooth (n+l)-dimensional Riemannian manifold. We show that if V is an area-stationary union of three or more C1,μn-dimensional submanifolds-with-boundaryMk⊂N with a common boundary Γ, then Γ is smooth and each Mk is smooth up to Γ (real-analytic in the case N is real-analytic). This extends a previous result of the author for codimension l=1. We additionally show that if {Vt}t∈(−1,1) is a Brakke flow such that each time-slice Vt is a union of three or more n-dimensional submanifolds-with-boundary Mk,t⊂N with a common boundary Γt and with parabolic C2+μ regularity in time-space, then {Γt}t∈(−1,1) and {Mk,t}t∈(−1,1) are smooth (second Gevrey with real-analytic time-slices in the case N is real-analytic).
We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, n≥3.
We show the uniqueness of strictly convex closed smooth self-similar solutions to the α-Gauss curvature flow with (1/n)<α<1+(1/n). We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the α-Gauss curvature flow with (1/n)<α<1+(1/n) shrinks a strictly convex closed smooth hypersurface to a round sphere.
This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Krö16]. Up to a small gap in the case of sin-cones, the stability properties of such warped products are now completely determined by spectral properties of the Laplacian and the Einstein operator of the fiber. For cosh-cylinders, we are furthermore able to prove a convergence result for the Ricci flow starting in a small neighbourhood. As an interesting class of examples, we determine the stability properties of sin-cones over symmetric spaces of compact type.
In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.
In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to O(n)-invariant cones are rotationally symmetric.
In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when n=2, the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, the weak minimizer of the Mabuchi energy is a smooth one.
We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is not available. Second, we prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists for all time and converges pointwisely to a hyperbolic metric
We consider the mean curvature flow of the graph of a smooth map f:R2→R2 between two-dimensional Euclidean spaces. If f satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map ft. Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map ft.
Let π:P(O(0)⊕O(k))→Pn−1 be a projective bundle over Pn−1 with 1≤k≤n−1. In this paper, we show that lens space L(k;1)(r) with radius r embedded in P(O(0)⊕O(k)) is a self-similar solution, where P(O(0)⊕O(k)) is endowed with the U(n)-invariant gradient shrinking Ricci soliton structure. We also prove that there exists a pair of critical radii r1<r2 which satisfies the following. The lens space L(k;1)(r) is a self-shrinker if r<r2 and self-expander if r2<r, and the Ricci-mean curvature flow emanating from L(k;1)(r) collapses to the zero section of π if r<r1 and to the ∞-section of π if r1<r. This gives explicit examples of Ricci-mean curvature flows.
We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even Lp-Minkowski problems in Rn+1 for p>−n−1.
We prove a criterion for K-stability of a Q-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Székelyhidi, it yields a criterion for the existence of a Kähler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of Kähler-Ricci solitons.
Here we provide uniqueness of vanishing viscosity solutions to sub-Riemannian mean curvature flow problem, which was known only far from characteristic points or under special symmetry condition. We employ vanishing viscosity approach and look for solutions as limit of solutions to approximating flow, which is well defined also at characteristic points, and estimate the rate of convergence of the approximating solutions. The results are provided in the settings of both 3-dimensional rototranslation group SE(2) and Heisenberg group H1 and they are particularly important due to their relation to surface completion problem of model of the visual cortex.
With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula dtdRenV(Mn,g(t))=−TR∫Mn(S(g(t))+n(n−1))dVg(t), where S(g(t)) is the scalar curvature for the evolving metric g(t), and TR∫(⋅)dVg is Riesz renormalization. This extends our earlier work to a broader class of metrics.
We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.
We define Type I singularities for the mean curvature flow associated to a density ψ (ψMCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the ψ-curvature due to the density. We describe a family of curves whose evolution under ψMCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.
In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension n>2, there exists an embedded surface in Rn evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When n>3 this result generalizes the analogue result of Grayson for the classical mean curvature flow. Interestingly, when n=2, our result provides instead a counterexample in the nonlocal framework to the well known Grayson Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point.
Self-shrinkers are the special solutions of mean curvature flow in Rn+1 that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow. In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White, namely that any closed hypersurface in Rn+1 has entropy at least that of the round sphere, holds in any dimension n. This result had previously been established for the cases n≤6 by Bernstein-Wang using a carefully constructed weak flow. The main technical result of this paper is an extension of Colding-Minicozzi's classification of entropy-stable self-shrinkers to the singular setting. In particular, we show that any entropy-stable self-shrinker whose singular set satisfies Wickramasekera's α-structural hypothesis must be a round cylinder Sk(2k)×Rn−k.
The linear stability of warped productEinstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein metrics are unstable. By exploiting the relationship between warped product Einstein metrics, quasi-Einstein metrics and Ricci solitons, we introduce a new destabilising perturbation (the Ricci variation) and show that certain infinite families of warped product Einstein metrics will be unstable in high dimensions.
We prove that the moduli space of 2-convex embedded n-spheres in R^n+1 is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.
In this work we give a detailed description of Matthias Günther's proof of the Isometric Embedding Theorem of Riemannian manifolds. Subsequently we will use this method to show that it is possible to construct an isometric embedding of a geometric flow, for instance the Ricci-flow, into some Euclidean space.
We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any n∈N0 and for any negative line bundle L over a compact Kähler manifold D, the total space of the vector bundle L⊕(n+1) admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if c1(KD⊗(L∗)⊗(n+1))>0. This generalises the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler-Ricci solitons on Cn with positive curvature operator on (1,1)-forms is path-connected.
We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flowωt, which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time T<∞. We prove that the scalar curvature of ωt is bounded from above by C/(T−t)2 under the existence of a contraction associated to the limiting cohomology class [ωT]. This generalizes Z. Zhang's work to the conic case.
In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow.
We show that the results in are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a metric with zero curvature. We also give some results about the range of discrete Gaussian curvatures, which generalize Andreev-Thurston's theorem to some extent.
We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equation for their metrics yields a system of partial differential equations. Solving such system, we prove that almost all the four-dimensional pseudo-Riemannian generalized symmetric spaces of type B are Ricci solitons. These examples show some deep differences arising for the Ricci soliton equation between the Riemannian and the pseudo-Riemannian cases, as any homogeneous Riemannian Ricci soliton is algebraic [21]. We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.
The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an adapted form of the Ricci flow.
The purpose of the present article is to study and characterize sev- eral types of symmetries of generalized Robertson-Walker space-times. Con- formal vector fields, curvature and Ricci collineations are studied. Many im- plications for existence of these symmetries on generalied Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized Robertson- Walker space-times admitting conformal vector fields are investigated.
We prove that the Ricci flow that contracts a hyperbolic cusp has curvature decay like one over time squared. In order to do this, we prove a new Li-Yau type differential Harnack inequality for Ricci flow on surfaces.
We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and a volume constraint.
Models of geometric flows pertaining to R2 scale invariant (super) gravity theories coupled to conformally invariant matter fields are investigated. Related to this work are supersymmetric scalar manifolds that are isomorphic to the Kählerian spaces Mn=SU(1,1+k)/U(1)×SU(1+k) as generalizations of the non-supersymmetric analogs with SO(1,1+k)/SO(1+k) manifolds. For curved superspaces with geometric evolution of physical objects, a complete supersymmetric theory has to be elaborated on nonholonomic (super) manifolds and bundles determined by non-integrable superdistributions with additional constraints on (super) field dynamics and geometric evolution equations. We also consider generalizations of Perelman's functionals using such nonholonomic variables which result in the decoupling of geometric flow equations and Ricci soliton equations with supergravity modifications of the R2 gravity theory. As such, it is possible to construct exact non-homogeneous and locally anisotropic cosmological solutions for various types of (super) gravity theories modelled as modified Ricci soliton configurations. Such solutions are defined by employing the general ansatz encompassing coefficients of generic off-diagonal metrics and generalized connections that depend generically on all spacetime coordinates. We consider nonholonomic constraints resulting in diagonal homogeneous configurations encoding contributions from possible nonlinear parametric geometric evolution scenarios, off-diagonal interactions and anisotropic polarization/ modification of physical constants. We also analyze the conditions under which such modified mimetic type theories and solutions reproduce the Starobinsky inflationary models in the double scalar approach.
In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigenvalue problems, which is sharp in the sense of the order of the eigenvalues. As some interesting applications, we study the eigenvalue of drifting Laplacian on Ricci solitons, self-shrinkers and product Riemannian manifolds. We give the explicit upper bounds of the gap of the consecutive eigenvalues of the drifting Laplacian. Since eigenvalues is invariant in the sense of isometry, by the classifications of Ricci solitons and self-shrinkers, we give the explicit upper bounds for the consecutive eigenvalues of the drifting Laplacian on a large class metric measure spaces. In addition, we also consider the case of product Riemannian manifolds with certain curvature conditions and some upper bounds are obtained. Basing on the case of Laplace operator, we also present a conjecture as follows: all of the eigenvalues of the Dirichlet problem of drifting Laplacian on metric measure spaces satisfy: λk+1−λk≤(λ2−λ1)kn1.We note the conjecture is true in some special cases.
In this paper, we study the mean curvature flow of graphs with Neumann boundary condition. The main aim is to use the maximum principle to get the boundary gradient estimate for solutions. In particular, we obtain the corresponding existence theorem for the mean curvature flow of graphs.
In this paper, we study solitons on 3-dimensional manifolds. In particular, we show that 3-dimensional pseudo-symmetric gradient Ricci solitons and nontrivial gradient Yamabe solitons are locally isometric to either R3, S3, H3, R×S2 or R×H2.
The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the non-Kähler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov–Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric. In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing.
On a polarized manifold (X,L), the Bergman iteration φk(m) is defined as a sequence of Bergman metrics on L with two integer parameters k,m. We study the relation between the Kähler-Ricci flowφt at any time t≥0 and the limiting behavior of metrics φk(m) when m=m(k) and the ratio m/k approaches to t as k→∞. Mainly, three settings are investigated: the case when L is a general polarization on a Calabi-Yau manifold X and the case when L=±KX is the (anti-) canonical bundle. Recently, Berman showed that the convergence φk(m)→φt holds in the C0-topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman's result and show that this convergence actually holds in the smooth topology.
Let n≥3 and m=n+2n−2. We construct 5-parameters, 4-parameters, 3-parameters ancient solutions of the equation vt=(vm)xx+v−vm, v>0, in R×(−∞,T) for some T∈R. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as ∣x∣→∞. We also prove that both the 3-parameters ancient solution and the 4-parameters ancient solution are singular limit solution of the 5-parameters ancient solutions.
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian cycles (possibly with differing phases). We use the flow of cycles to obtain the fact that special lagrangian cycles generate the part of the lagrangian homology which lies in the image of the Hurewicz homomorphism. We also establish a weak version of a conjecture of Thomas-Yau regarding lagrangian mean curvature flow.
The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder R×Sn−1/Γ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity. The result is then applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It is previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of R×S3 or converges to a quotient of R2×S2 along each integral curve of the gradient vector field of the potential function.
In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.
We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of FanoKähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of κ-collapsed and κ-noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions ≥7. In dimension 7 such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of κ-collapsed ancient solutions of type I are also obtained on torus bundles for all dimensions ≥8.
We show that any strictly mean convex translator of dimension n≥3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n≥3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of two-convex hypersurfaces by the two-harmonic mean curvature.
We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the {\it original manifold}. These results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead. Applications on convergence and degeneration of Riemannian metrics under this curvature condition are given. To this effect, in particular for the Bakry-Émery Ricci curvature condition, the gradient of the potential function is allowed to have singularity of order close to 1 while the traditional method of weighted manifolds allows bounded gradient. This approach enables us to extend some of the results in the papers,,, and. The condition also covers general Ricci solitons instead of just gradient Ricci solitons.
In this short note we observe that the concavity of Perelman's W-functional over a neighborhood of a Kähler-Ricci soliton inside the space of Kähler potentials is a direct consequence of author's solution of the variational stability problem for Kähler-Ricci solitons. Independently, we provide a rather simple proof of this fact based on some elementary formulas obtained in our previous work.
In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field on manifolds with bounded Ricci curvature.
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.
In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the time derivatives of the conformal factor are bounded and for each fixed time, the conformal factor has an explicit asymptotic expansion near the cone points.
We study a Boltzmann's type entropy functional (which appeared in existing literature) defined on Kähler metrics of a fixed Kähler class. The critical points of this functional are gradient Kähler-Ricci solitons, and the functional was known to be monotonically increasing along the Kähler-Ricci flow in the canonical class. In this article, we derive and analyze the second variation formula for this entropy functional, and show that all gradient Kähler-Ricci solitons are stable with respect to this entropy functional. Furthermore, using this result, we give a new proof that gradient shrinking Kähler-Ricci solitons are stable with respect to the Perelman's entropy in a fixed Kähler class.
In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in CPm. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t→∞. Consequently, we obtain a new differentiable sphere theorem for submanifolds in CPm. Our work improves the convergence theorem for mean curvature flow due to Pipoli and Sinestrari .
Mean curvature flow of clusters of n-dimensional surfaces in R^n+k that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and the real line. In particular, we prove that there exists a large class of closed initial surfaces, as geodesic graphs over the totally geodesic surface Σ, such that the mean curvature flow exists for all time and converges to Σ. This is among the first examples of converging mean curvature flows of compact hypersurfaces in Riemannian manifolds. We also provide some useful calculations for the general warped product setting.
We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.
This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov-Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.
We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant–Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.
In this paper we prove that a certain class of embedded unknotted curves in R3 evolving under curve shortening flow do not form singularities Type II before collapsing to a point. Our proof uses tools of the minimal surface theory to study a suitable isoperimetric ratio.
We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
We study the asymptotic Dirichlet problem for f-minimal graphs in Cartan-Hadamard manifolds M. f-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of f-minimal graphs with prescribed boundary behavior on a bounded domain Ω⊂M under suitable assumptions on f and the boundary of Ω. In the second part, we consider the asymptotic Dirichlet problem. Provided that f decays fast enough, we construct solutions to the problem. Our assumption on the decay of f is linked with the sectional curvatures of M. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let (Mn,g0) be a n-dimensional smooth compact manifold with boundary, where n≥3, assume the conformal invariant Y(M,∂M)<0. Given any negative smooth functions f in M and h on ∂M, there exists a unique conformal metric of g0 such that its scalar curvature equals f and mean curvature curvature equals h. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In the flow approach, assume another conformal invariant Q(M,\paM) is a negative real number, for some class of initial data, we prove the short time and long time existences of the so-called prescribed scalar curvature plus mean curvature flows, as well as their asymptotic convergence. Via a family of such flows together with some additional variational arguments, under the flow assumptions we prove existence and uniqueness of positive minimizers of the associated energy functional and also the above result by analyzing asymptotic limits of the flows and the relations among some conformal invariants.
Given a smooth, symmetric, homogeneous of degree one function f=f(λ1,⋯,λn) satisfying ∂if>0 for all i=1,⋯,n, and an oriented, properly embedded smooth cone Cn in Rn+1, we show that under some suitable conditions on f and the covariant derivatives of the second fundamental form of C, there is at most one fself-shrinker (i.e. an oriented hypersurface Σn in Rn+1 for which f(κ1,⋯,κn)+21X⋅N=0 holds, where X is the position vector, N is the unit normal vector, and κ1,⋯,κn are principal curvatures of Σ) that is asymptotic to the given cone C at infinity.
In, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant curvature PL-metric.
In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flow can be used to deform any inversive distance circle packing to a unique packing with prescribed cone angle. We also give partial results on the range of all admissible cone angles, which generalize the classical Andreev-Thurston's theorem.
We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions on noncompact Riemannian manifolds with at most quadratic curvature growth, whose Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method implies that, under some appropriate assumptions, the spatial transfer rate (the rate at which high curvature regions affect low curvature regions) of the Ricci flow resembles that of the heat equation.
We study geometric properties of the Lagrangian self-shrinking tori in R4. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in. When the area bound is small, we show that any Lagrangian self-shrinking torus in R4 with small area is embedded with uniform curvature estimates, and the space of such tori is compact. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R4, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in R3 in.
Let (X,ω) be a compact normal Kähler space, with Hodge metric ω. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on Hω(X), the space of Kähler potentials, whose completion is the finite energy space Eω1(X). Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of Kähler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy in a suitable sense. To our knowledge, this result represents the first characterization of general log Fano pairs admitting Kähler-Einstein metrics. We also discuss the analogous result for Kähler-Ricci solitons on Fano varieties.
In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let (Mn,g) be a compact gradient shrinking Ricci soliton satisfying Ricg+Ddf=ρg with ρ>0 constant. We show that if (M,g) satisfies δW(⋅,⋅,∇f)=0, then (M,g) is Einstein. Here W denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in, and.
Studying the behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler-Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models.
We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.
Let Δφ=Δ−∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ=e−φdv on an n-dimensional compact Riemannian manifold (M,g), where g=g(t) solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of Δφ and we obatin several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for the case n≥3. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.
In this short note, we give simple proof of the Ricci flow's local existence and uniqueness on closed Einstein manifolds. We suggest a new setting for studying the space of Riemannian metrics on a compact manifold.
In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our key ingredient is the existence result of general curvature flow with boundary conditions, which is proved in Section 4.
We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in Rn. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of the flowing surface onto Rn, the shadow of the flow. This moving shadow can be seen as a weak solution for mean curvature flow of hypersurfaces in Rn with a Dirichlet boundary condition. Furthermore, we provide a lemma of independent interest to locally mollify the boundary of an intersection of two smooth open sets in a way that respects curvature conditions.
In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold (M,g,J). We prove that if the initial metric has Griffiths positive (non-negative) Chern curvature Ω, then this property is preserved along the flow. On a manifold with Griffiths non-negative Chern curvature the HCF has nice regularization properties, in particular, for any t>0 the zero set of Ω(ξ,ξˉ,η,ηˉ) becomes invariant under certain torsion-twisted parallel transport.
We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneousRicci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also determined, and the energy of critical points is explicitly calculated.
On an n-dimensional complete manifoldM, consider an h-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and dh/du>0, then the manifold M is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of M is four, the metric g is conformally flat.
We show that the twisted Kähler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilton-Jacobi equations and hyperbolic conservation laws (notably, in the adhesion model in cosmology). In particular we show how to recover the Hele-Shaw flow (Laplacian growth) of growing 2D domains from the Ricci flow. As we briefly indicate the scaling limit in question arises as the zero-temperature limit of a certain many particle system on X.
In this article we study any 4-dimensional Riemannian manifold (M,g) with harmonic curvature which admits a smooth nonzero solution f to the following equation \begineqnarray \nabla df = f(Rc -\fracRn-1 g) + x Rc+ y(R) g. \endeqnarray where Rc is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvatureR. We show that a neighborhood of any point in some open dense subset of M is locally isometric to one of the following five types; {\rm (i)} S2(6R)×S2(3R) with R>0, {\rm (ii)} H2(6R)×H2(3R) with R<0, where S2(k) and H2(k) are the two-dimensional Riemannian manifold with constant sectional curvature k>0 and k<0, respectively, {\rm (iii)} the static spaces in Example 3 below, {\rm (iv)} conformally flat static spaces described in Kobayashi's, and {\rm (v)} a Ricci flat metric. We then get a number of Corollaries, including the classification of the following four dimensional spaces with harmonic curvature; static spaces, Miao-Tam critical metrics and V-static spaces. The proof is based on the argument from a preceding study of gradient Ricci solitons. Some Codazzi-tensor properties of Ricci tensor, which come from the harmonicity of curvature, are effectively used.
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0) form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.
We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons(M,g,f) which have harmonic Weyl curvature, i.e. δW=0. Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product R2×Nλ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature λ=0, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works and Derdziński's study on Codazzi tensors. Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with δW=0. For shrinking case, it reproves the rigidity result in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally-flat ones with δW=0. We also characterize {\it locally} 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.
We prove uniform curvature estimates for homogeneousRicci flows: For a solution defined on [0,t] the norm of the curvature tensor at time t is bounded by the maximum of C(n)/t and C(n)(scal(g(t))−scal(g(0))). This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension n. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local W2,p convergence result, which holds without symmetry assumptions.
In this paper we study the Ricci flowon surfaces homeomorphic to a cylinder (that is, a product of the circle with a compact interval). We prove longtime existence results, results on the asymptotic behavior of the flow, and we report on an interesting phenomenon: convergence to constant curvature in the normalised flow,under certain assumptions on the initial data, cannot be exponential.
Consider the Allen-Cahn equation on the d-dimensional torus, d=2,3, in the sharp interface limit. As it is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn equation and describe its large deviation asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviation upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events.
We consider the evolution of hypersurfaces on the unit sphere Sn+1 by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Alexandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.
First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type (other than a (real) hyperbolic space) and the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature over the invariant submanifold in the C∞-topology in infinite time.
In this note, we show that any n-dimensional κ-noncollapsed steady Kähler-Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in.
Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curvature on a large, but compact, set must be a round cylinder. Using this result, Colding and Minicozzi were able to establish uniqueness of blowups at cylindrical singularities, and provide a detailed description of the singular set of generic mean curvature flows. In this paper, we show that the bounded curvature assumption is unnecessary for the rigidity of the cylinder if either n is at most 6, or if the mean curvature is bounded below by a positive constant. These results follow from curvature estimates that we prove for strictly mean convex self-shrinkers. We also obtain a rigidity theorem in all dimensions for graphical self-shrinkers, and curvature estimates for translators of the mean curvature flow.
We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.
{We examine the critical behaviour i. e. P−V criticality of conformal gravity (CG) in an extended phase space in which the cosmological constant should be interpreted as a thermodynamic pressure and the corresponding conjugate quantity as a thermodynamic volume.} The main potential point of interest in CG is that there exists a non-trivial \emph{Rindler parameter (a)} in the {spacetime geometry. This geometric parameter has an important role to construct a model for gravity at large distances where the parameter "a" actually originates}. We also investigate the effect of the said parameter on the black hole (BH) thermodynamic equation of state, critical constants, Reverse Isoperimetric Inequality, first law of thermodynamics, Hawking-Page phase transition and Gibbs free energy for this BH. We speculate that due to the presence of the said parameter, there has been a deformation in the shape of the isotherms in the P−V diagram in comparison with the charged-AdS (anti de-Sitter) BH and the chargeless-AdS BH. Interestingly, we find that the critical ratio for this BH is ρc=TcPcvc=23(32−23), which is greater than the charged AdS BH and Schwarzschild-AdS BH i.e. ρcCG:ρcSch−AdS:ρcRN−AdS=0.67:0.50:0.37. The symbols are defined in the main work. Moreover, we observe that {the critical ratio has a constant value} and it is independent of the non-trivial \emph{Rindler parameter (a)}. Finally, we derive the reduced equation of state in terms of the reduced temperature, the reduced volume and the reduced pressure respectively.
We show the consistency of a threshold dynamics type algorithm for the anisotropic motion by fractional mean curvature, in the presence of a time dependent forcing term. Beside the consistency result, we show that convex sets remain convex during the evolution, and the evolution of a bounded convex set is uniquely defined.
In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat incomplete edge metric, we prove long time existence of the Ricci de Turck flow. Under certain conditions, our results yield existence of Ricci flow on spaces with incomplete edge singularities. The proof works by a careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.
We introduce a notion of index for shrinkers of the mean curvature flow. We then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in R^3, namely, that such shrinkers have index at least 3, unless they are one of the stable ones: the sphere, the cylinder, or the plane. We also provide a generalization of the result to higher dimensions.
In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph (y,u(∣y∣)) over Rn, n≥2, satisfying there exist positive constants ε, c and N such that u′(r)≥crε for r≥N, the longtime solution to mean curvature flow with initial data (y,u(∣y∣)) must be Type IIb. We also study the asymptotic behavior of Type IIb mean curvature flow and show that the limit of suitable rescaling sequence for mean-convex Type IIb mean curvature flow satisfying δ-Andrews' noncollapsing condition is translating soliton.
In this paper, we prove that if an asymptotically Euclidean manifold (Mn,g) under the condition that R≥0 has long time existence of Ricci flow, the mass of (Mn,g) is nonnegative. In addition, we give an independent proof of positive mass theorem in dimension 3.
We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional assumptions in [13] by Wallach and the second author.
The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan-Hadamard spaces as well as on minimal submanifolds of Rn. The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with non-negative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean. We then pass to consider the problem of the existence and the regularity of optimal shapes (i.e. regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. We give examples of spaces where there exists no isoperimetric-isodiametric region and we prove that on the other hand on compact spaces and on C0-locally-asymptotic Euclidean manifolds with non-negative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this last class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so called Hawking gravitational instantons and the Bryant-type Ricci solitons). We finally prove that the boundary of the isoperimetric-isodiametric regions satisfy the optimal C1,1 regularity outside of a closed subset of Hausdorff co-dimension 8.
In this paper we generalize the theory of Cheeger, Colding and Naber to certain singular spaces that arise as limits of sequences of Riemannian manifolds. This theory will have applications in the analysis of Ricci flows of bounded curvature, which we will describe in a subsequent paper.
In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension ≥4. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension ≥4. In the course of the proof, we will also establish Lp<2-curvature bounds on time-slices of such flows.
We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane C2. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in C2 such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.
Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each type of curvature, with the integrals based on the parallel transport of vectors around hinges. Computations for triangulations of a diverse set of manifolds show these piecewise flat curvatures to converge to their smooth values. The Ricci curvature also gives a piecewise flat Ricci flow as a fractional rate of change of edge-lengths, again converging to the smooth Ricci flow for the manifolds tested.
We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient v3 in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are stable with respect to the associated W-functional.
We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces (X,dt,mt)t∈I. The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of the synthetic lower Ricci bounds of Lott-Sturm-Villani for static metric measure spaces, the defining property for super-Ricci flows is the `dynamic convexity' of the Boltzmann entropy Ent(.∣mt) regarded as a functions on the time-dependent geodesic space (P(X),Wt)t∈I. For Ricci flows, in addition a nearly dynamic concavity of the Boltzmann entropy is requested. Alternatively, super-Ricci flows will be studied in the framework of the Γ-calculus of Bakry-Émery-Ledoux and equivalence to gradient estimates will be derived. For both notions of super-Ricci flows, also enforced versions involving an `upper dimension bound' N will be presented.
We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the Zoll Finsler metrics along the retraction.
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetricnoncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ>1/2, there is a solution with the highest curvature blowing up at the rate (T−t)−(γ+1/2). (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.
Bence-Merriman-Osher algorithm computes numerically mean curvature flow via solutions of heat equation iteratively initialized at the end of each short time interval. Inspired by the convergence proof of Evans of this algorithm where he employed nonlinear semigroup theory and level set approach to mean curvature flow, we prove here that a similar procedure, but now with a different definition of evolved surface at the end of each short time interval, results in a motion by curvature. The interest of this version of the algorithm is its application corresponding to model of the visual cortex.
We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in the case of some Euclidean cones, a singularity persists at the apex. Secondly, we generalize the construction to a sub-Riemannian manifold, namely the Heisenberg group, and show that it regularizes the space instantaneously to a smooth Riemannian manifold.
In this paper, we study elliptic gradient estimates for a nonlinear f-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear f-heat equation only assuming the Bakry-Émery Ricci tensor is bounded below. As applications, we prove parabolic Liouville properties for some kind of ancient solutions to the nonlinear f-heat equation. Some special cases are also discussed.
We use the bracket flow/algebraic soliton approach to study the Laplacian flow of G2-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a G-invariant G2-structure on a homogeneous space G/K that flows by pull-back of automorphisms of G up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space CP2n+1=Sp(n+1)/(U(1)×Sp(n)).
In this article we give a complete description of the evolution of an area decreasing map f:M→N induced by its mean curvature in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures σM, σN satisfy minσM≥supσN.
We study geometric relativistic flow and Ricci soliton equations which (for respective nonholonomic constraints and self-similarity conditions) are equivalent to the gravitational field equations of R2gravity and/or to the Einstein equations with scalar field in general relativity, GR. Perelman's functionals are generalized for modified gravity theories, MGTs, which allows to formulate an analogous statistical thermodynamics for geometric flows and Ricci solitons. There are constructed and analyzed generic off-diagonal black ellipsoid, black hole and solitonic exact solutions in MGTs and GR encoding geometric flow evolution scenarios and nonlinear parametric interactions. Such new classes of solutions in MGTs can be with polarized and/or running constants, nonholonomically deformed horizons and/or imbedded self-consistently into solitonic backgrounds. They exist also in GR as generic off-diagonal vacuum configurations with effective cosmological constant and/or mimicking effective scalar field interactions. Finally, we compute Perelman's energy and entropy for black ellipsoids and evolution solitons in R2 gravity.
We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman, Bence and Osher. We prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation of mean curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movement scheme by Esedoglu et. al.. This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movement scheme for an energy functional that Γ-converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren, Taylor and Wang and Luckhaus and Sturzenhecker, which establish convergence of a more academic minimizing movement scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which however is not ensured by the compactness coming from the basic estimates.
By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface Mn which evolves under inverse mean curvature flow in Rn+1, the isoperimetric lower bounds for both eigenvalues were founded.
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.
Suppose (X,ω) is a compact Kähler manifold. We introduce and explore the metric geometry of the Lp,q-Calabi Finsler structure on the space of Kähler metrics H. After noticing that the Lp,q-Calabi and Lp′-Mabuchi path length topologies on H do not typically dominate each other, we focus on the finite entropy space EEnt, contained in the intersection of the Lp-Calabi and L1-Mabuchi completions of H and find that after a natural strengthening, the Lp-Calabi and L1-Mabuchi topologies coincide on EEnt. As applications to our results, we give new convergence results for the Kähler–Ricci flow and the weak Calabi flow.
We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.
We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White to the free-boundary setting. We use Ilmanen's elliptic regularization procedure to prove existence of free-boundary Brakke flows.
In this paper, we study the motion of level sets by general curvature. The difficulty of this setting is that a general curvature function is only well defined in an admissible cone. In order to extend the existence of a weak solution of a general curvature flow to outside the cone we introduce a new approximation function f^n (see (3.1)). Moreover, using the idea in [4], we give an elliptic approach for the Ben-Andrews' non-collapsing result in fully nonlinear curvature flows; we hope this approach can be generalized to a wider class of elliptic equations.
We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all n-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with ∣Ric∣≤K must satisfy ∣Rm∣≤Ct−1 for all t∈(0,T], where C depends only on the dimension n and T depends on K and the injectivity radius injg(t). In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown. In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.
Warped product metrics are a class of Riemannian metrics on cross products B×F which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base B has two dimensions, the metric must in fact be a cross product. The theorem is a consequence of a Pohozaev-type identity. This closely parallels results in the study of blow-ups of solutions to the reaction- diffusion equation ∂tv=Δv+ev in R2 .
Given a sequence of asymptotically flat3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed C0 Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken's isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken's work before explaining our extension of that work. The results are all specific to three dimensions.
We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any n-dimensional (n≥4) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of Nn−k×Rk, (k>0), the product of a Einstein manifoldNn−k with the Gaussian shrinking soliton Rk. The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton.
We prove that every entire self-shrinking solution on Cn to the Kähler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtained by Q. Ding and Y.L. Xin. Furthermore, we show that our argument works for a larger class of equations.
We use the time real analyticity of Ricci flow proved by Kotschwar to extend a result in , namely, we prove that the Laplace spectra of negatively curved compact surfaces having same genus γ≥2, same area and same curvature bounds vary in a "controlled way", of which we give a quantitative estimate (Theorem 1.1 below). We also observe how said real analyticity can lead to unexpected conclusions about spectral properties of generic metrics on a compact surface of genus γ≥2 (Proposition 1.5 below).
In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in Rm+1. We also obtain an upper bound to the maximum height that a compact embedded translator in R3 can achieve. On the other hand, we study graphical perturbations of translators, showing that asymptotic graphical perturbations of a graph translator of revolution remain a hypersurface of revolution. Finally, we prove that compact translators that lie between two parallel planes inherit the symmetries of their boundaries curves.
We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time L2-bound for the time derivative of the solution. The result is surprising when compared with results for the harmonic map heat flow, where non-uniqueness through reverse bubbling may occur.
In this paper we prove that if γ is a Jordan curve on S2 then there is a smooth curve shortening flow defined on (0,T) which converges to γ in C0 as t→0+. Another perspective is that the level-set flow of γ is smooth. This is a generalization of the author's previous work where the planar case was studied. If a Jordan curve on S2 has Lebesgue measure zero then we show that the level-set flow instantly becomes a smooth closed curve. If the Lebesgue measure is positive then for small time the level-set flow is an annulus with smooth boundary. This second case should be interpreted as a failure of uniqueness. As in the planar case a key step in the proof is establishing a length estimate for smooth curves that depends on a geometric quantity called the r-multiplicity. The majority of this paper concerns the extension of this length estimate to S2.
In this paper we discuss the stability of geodesic spheres in Sn+1 under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold setting.
We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing term tends to infinity in a certain manner. This result allows the Morse homology to be explicitely calculated, and will permit us to show in forthcoming work that, for a large family of smooth positive functions, F, defined over a (d+1)-dimensional flat torus, there exist at least 2d+1 distinct, locally strictly convex, Alexandrov-embedded hyperspheres of mean curvature prescribed at every point by F.
We study the generalized Kähler-Ricci flow on complex surfaces with nondegenerate Poisson structure, proving long time existence and convergence of the flow to a weak hyperKähler structure.
In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ε-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn,g,f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvatureR(x) satisfies limr(x)→∞R(x)f(x)=C0supx∈MR(x) with C0>2n−2.
We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.
In this paper, we prove the long-time existence and uniqueness of the conical Kähler-Ricci flow with weak initial data which admits Lp density for some p>1 on Fano manifold. Furthermore, we study the convergence behavior of this kind of flow.