Papers from 2021

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins-Sekerka flow.

We find examples of cohomogeneity one metrics on $S^4$ and $\mathbb{C}{P}^2$ with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

In a Riemannian manifold, the Ricci flow is a partial differential equation for evolving the metric to become more regular. We hope that topological structures from such metrics may be used to assist in the tasks of machine learning. However, this part of the work is still missing. In this paper, we propose Ricci flow assisted Eucl2Hyp2Eucl neural networks that bridge this gap between the Ricci flow and deep neural networks by mapping neural manifolds from the Euclidean space to the dynamically stable Poincaré ball and then back to the Euclidean space. As a result, we prove that, if initial metrics have an $L^2$-norm perturbation which deviates from the Hyperbolic metric on the Poincaré ball, the scaled Ricci-DeTurck flow of such metrics smoothly and exponentially converges to the Hyperbolic metric. Specifically, the role of the Ricci flow is to serve as naturally evolving to the stable Poincaré ball. For such dynamically stable neural manifolds under the Ricci flow, the convergence of neural networks embedded with such manifolds is not susceptible to perturbations. And we show that Ricci flow assisted Eucl2Hyp2Eucl neural networks outperform with their all Euclidean counterparts on image classification tasks.

We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles $\alpha \in (0,\pi )$. First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess.

The mathematical model of grain boundary motion, including lattice misorientations' effect, is considered. When time-dependent lattice misorientations are state variables of the surface tension of the grain boundary, to ensure the energy dissipation law, one can obtain a curvature flow of networks with time-dependent mobilities. This paper studies the solvability and long-time asymptotic behavior of the curvature flow subjected to the Herring condition which ensures that the constituent grain boundary surface tensions are balanced at the triple junction.

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\le cR$ for some positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for $4$-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a $3$-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Chodosh 14 in dimension three and show that any $3$-dimensional gradient Ricci expander $C^2$ asymptotic to $\left(C({\mathbb{S}}^2),d{t}^2+\alpha t^2{g}_{{\mathbb{S}}^2}\right)$ is rotationally symmetric, where $\alpha \in (0,1]$ is a constant and $g_{{\mathbb{S}}^2}$ is the standard metric on ${\mathbb{S}}^2$ with constant curvature $1$.

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to address these questions, partial results of Boehm, Lafuente, and Lauret are extended to show that left-invariant Ricci solitons on solvable Lie groups are unique up to scaling and isometry.

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

A point source of light is placed inside an oval. The $n$-th caustic by reflection is the envelope of the light rays emanating from the light source after $n$ reflections off the curve. We show that each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement, concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

In this paper, we prove a pseudolocality-type theorem for $\mathcal{L}$-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the $\mathcal{L}$-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L. Chen.

Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $\sigma$-models. We include non-conformal $\sigma$-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General $\sigma$-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry. This thesis is based on the conjecture that integrable $\sigma$-models are renormalizable, or stable under the RG flow. It is widely understood that classically integrable theories are stable at the leading 1-loop order with only a few parameters running. Here we address what happens at higher-loop orders. We find that integrable $\sigma$-models generally remain RG-stable at higher-loops provided they receive a particular choice of finite counterterms, or quantum (${\alpha }^{\prime }$) corrections to the target-space geometry. We explicitly construct these quantum corrections for examples of integrable $\eta$- and $\lambda$-deformed $\sigma$-models. We then reformulate the $\lambda$-models as $\sigma$-models on a "tripled" $G\times G\times G$ configuration space, where they become automatically renormalizable due to manifest symmetries and a decoupling of some fields. We also consider the integrable $G\times G$ and $G\times G/H$ models and construct a new class of integrable $G\times G/H$ models with abelian $H$. We then present a new and different link between integrability and the RG flow in the context of $\sigma$-models with 'local couplings' depending explicitly on 2d time. Such models are naturally obtained in the light-cone gauge in string theory, pointing to the possibility of a large, new class of solvable string models.

In this work we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a $1+1+2$ spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor with respect to the induced metric. These are slices of constant time in the spacetime. Firstly, the form of the Ricci tensor for general hypersurfaces is obtained and the conditions under which the general case reduces to those of constant time slices are specified. We provide a characterization of these hypersurfaces, with key physical quantities in the spacetime playing a role in specifying the local geometry of these hypersurfaces. Furthermore, we investigate the case where these hypersurfaces admit a Ricci soliton structure. The particular cases where the vector fields associated to the solitons are Killing or conformal Killing vector fields are analyzed. Finally, in the context of spacetimes with local rotational symmetry it is shown that, only spacetimes in this class with vanishing rotation and spatial twist can admit the hypersurface types considered, and that the hypersurfaces are necessarily flat. And if such hypersurface do admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.

In this paper, we study the forced mean curvature flows and the prescribed mean curvature equations of both graphs and level-sets with capillary-type boundary conditions on a $C^3$ bounded domain, which is not necessarily convex. We prove a priori gradient estimates locally Lipschitz in time. Under an assumption on the forcing term, we prove that the gradient estimates are globally Lipschitz in time. As a consequence, we obtain the existence theorem of solutions. In our formulation, we recover the known results of the gradient estimates on a strictly convex $C^3$ bounded domain. Next, we study the associated eigenvalue problems for mean curvature flows of both graphs and level-sets. We prove the large time behavior of the solutions of mean curvature flows of graphs on a smooth bounded domain. Finally, we compute the asymptotic speed of the solutions of level-set mean curvature flows and the large time profile of level-sets in the radially symmetric case based on optimal control formula. Examples arising in the radially symmetric case demonstrate that the additional assumption on the forcing term is optimal.

In this paper, we extend Tosatti's method to study the lower boundedness of modified $K$-energy on a Fano manifold and apply this result to study the relative $K$-stability of the deformation space of a Kähler Ricci soliton.

We follow a new pathway to the definition of the Stochastic Quantization (SQ), first proposed by Parisi and Wu, of the action functional yielding the Einstein equations. Hinging on the functional similarities between the Ricci-Flow equation and the SQ Langevin equations proposed by Rumpf, we push forward a novel approach characterized by a multiplicative noise and a stochastic time that converges to the proper time of a space-like foliation in the equilibrium limit, where quantities have constant averages. We express the starting system of equations using the Arnowitt-Deser-Misner (ADM) variables and their conjugated Hamiltonian momenta. Such a choice is instrumental in understanding the newly derived equations in terms of the breakdown of the diffeomorphism invariance of the classical theory, which instead will hold on average at the steady state. We comment on the physical interpretation of the Ricci flow equations, and argue how they can naturally provide, in a geometrical way, the renormalization group equation for gravity theories. In the general setting, the equation associated to the shift vector yields the Navier-Stokes equation with a stochastic source. Moreover, we show that the fluctuations of the metric tensor components around the equilibrium configurations, far away from the horizon of a Schwarzschild black hole, are forced by the Ricci flow to follow the Kardar-Parisi-Zhang equation, whose probabilistic distribution can yield an intermittent statistics. We finally comment on the possible applications of this novel scenario to the cosmological constant, arguing that the Ricci flow may provide a solution to the Hubble tension, as a macroscopic effect of scale dependence of the quantum fluctuations of the metric tensor.

A framework of quantum spacetime reference frame is proposed and reviewed, in which the quantum spacetime at the Gaussian approximation is deformed by the Ricci flow. At sufficient large scale, the Ricci flow not only smooths out local small irregularities making the universe a homogeneous and isotropic Friedmann-Robertson-Walker metric, but also develops a local singularity at the physical-time origin. Due to the phenomenological suppression of the non-Gaussian primordial perturbations, we assume the validity of the Ricci flow applying to the high curvature region near the local singularity of the early universe. The no-local-collapsing theorem of Perelman ensures the existence of a canonical neighborhood around the large curvature pinching point, which resembles a gradient shrinking Ricci soliton (GSRS) solution of the Ricci flow. Without any inflaton field, the GSRS naturally reproduces an exact inflationary deSitter universe near the singularity at the leading order. Without any rolling-down behavior of inflaton, the deviation from exact deSitter described by the "slow roll parameters" can be calculated by a small deviation from the singular flow-time via the Ricci flow, and the primordial perturbations can also be studied on the GSRS background, the power spectrum of the scalar perturbation agrees with present observations, and the one of the tensor perturbation is predicted too small to be detectable than the standard inflation. The previous treatment of the cosmological constant and the effective gravity are also briefly reviewed in the framework. So we argue that the Ricci flow provides us a possible unified view and treatment of the late epoch accelerating expansion and early epoch inflation of the universe without introducing dark energy or inflaton (dark energy of the second kind).

Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T-duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.

In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.

Using center manifolds and topological degree theory, we construct a new family of complete, $SU(2)$-invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram ${\mathbb{Z}}_4\subset U(1)\subset SU(2)$. We also provide simpler constructions of the existing $U(2)$-invariant steady and complete gradient solitons on the cohomogeneity one manifolds with group diagrams ${\mathbb{Z}}_n\subset U(1)\subset SU(2)$ for any $n\in \mathbb{N}$, including Appleton's non-collapsed solitons for $n\ge 3$.

In this paper, we consider $n$-dimensional compact K$\ddot{a}$hler manifold with semi-ample canonical line bundle under the long time solution of K$\ddot{a}$hler Ricci Flow. In particular, if the Kodaira dimension is one, Ricci curvature converge to negative of generalized K$\ddot{a}$hler Einstein metric ${\omega }_B$ locally away from singular set in $C_{loc}^0(\omega (t))$ topology.

In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space ${\mathbb{H}}^{n+1}$ with speed $(\sinh r{)}^{\alpha /\beta }{\sigma }_k^{1/\beta }$, where ${\sigma }_k$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $\alpha$, $\beta$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $\alpha$ and $\beta$. When $k=1,\alpha >1+\beta$, and the initial hypersurface is mean convex, we prove that the mean convex solution to the flow for $k=1$ exists for all time and converges smoothly to a sphere. When $1\le k\le n,\alpha >k+\beta$, and the initial hypersurface is uniformly convex, we prove that the uniformly convex solution to the flow exists for all time and converges smoothly to a sphere. In particular, we generalize Li-Sheng-Wang's results from Euclidean space to hyperbolic space.

We study the rescaled mean curvature flow (MCF) of hypersurfaces that are global graphs over a fixed cylinder of arbitrary dimensions. We construct an explicit stable manifold for the rescaled MCF of finite codimensions in a suitable configuration space. For any initial hypersurface from this stable manifold, we construct a unique global solution to the rescaled MCF, and derive precise asymptotics for these solutions that are valid for all time. Using these asymptotics, we prove asymptotic stability of cylindrical singularities of arbitrary dimensions under generic initial perturbations. As a by-product, for any flow of hypersurfaces evolving according to the MCF that enters this stable manifold at any time and first develops a singularity at a subsequent time, we give a simple proof of the uniqueness of tangent flow, first established by Colding and Minicozzi. Moreover, in this case we show the unique singularity profile is determined by the hypersurface profile when the flow enters the stable manifold. For all results in this paper, there is no symmetry or solitonic assumption.

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and its covariant derivative $\nabla Rm$ can be bounded by the scalar curvature $R$ by $|Rm|\le C_a{R}^a$ and $|\nabla Rm|\le C_a{R}^a$ (on $M\backslash K$), for any $0\le a<1$ and some constant $C_a>0$. Moreover, if the scalar curvature has at most polynomial decay at infinity, then $|Rm|\le CR$ (on $M\backslash K$). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton $(M^4,g,f)$ has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and $C^{1,\alpha }$ asymptotic cones at infinity ($0<\alpha <1$) according to Chen-Deruelle [20].

We obtain curvature estimates for long time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in arXiv:1410.3157 and arXiv:0709.0990, we adapt arguments from arXiv:1903.05939 for the Kähler-Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds.

We classify convex ancient curve shortening flows in the disc with free boundary on the circle.

We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t}=(r_{\phi }^m-R_{\phi }^m)g$, $\frac{\partial \phi }{\partial t}=\frac{m}{2}(R_{\phi }^m-r_{\phi }^m)$ on a smooth metric measure space $(M^n,g,e^{-\phi }{\mathrm{dvol}}_g,m)$, where $R_{\phi }^m$ denotes the associated weighted scalar curvature, and $r_{\phi }^m$ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension $n$ satisfies $n⩾3$.

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball ${\mathbb{B}}^{n+1}$ and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the $n$-th quermassintegral and non-decreases the $k$-th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in ${\mathbb{B}}^{n+1}$. This generalizes the result in for convex hypersurfaces with free boundary in ${\mathbb{B}}^{n+1}$.

In this article, we prove a local Sobolev inequality for complete Ricci flows. Our main result is that the local $\nu$-functional of a disk on a Ricci flow depends only on the Nash entropy based at the center of the disk, and consequently depends only on the volume of the disk. Furthermore, we introduce some applications of this local Sobolev inequality. These applications reveal the way in which the local geometry evolves along Ricci flow. In particular, we show that several classical theorems related to Perelman's monotonicity formula can be derived from our results.

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a sphere in smooth topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in (n+1) dimensional Euclidean space, but we also get the necessary and sufficient condition for the establishment of the equality. In the second part of this paper, we study a mean curvature type flow (1.7) of static convex hypersurfaces in (n+1) dimensional Euclidean space, and prove that the flow (1.7) has a unique smooth solution for all time t>0, and the static convexity of the hypersurface is preserved along the flow (1.7). Moreover, The solution of the flow (1.7) converges exponentially to a sphere of radius R in smooth topology as time tends to infinity. By exploiting the properties of this flow, we develop and present a new sharp Michael-Simon inequality for kth mean curvature.

We show that a complete gradient Ricci soliton $(M^n,g)$ with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a Kähler gradient Ricci soliton of real dimension $\ge 4$, with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar curvature.

In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is $W^{1,p}$ for some $n<p\le \infty$. As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in distributional sense, then the manifold is isometric to a Ricci flat manifold.

In this paper we study the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the general $n$- dimensional case, we prove uniqueness under additional conditions: we require a { \em uniform lower bound } on the second fundamental form and the height function of the initial condition. The latter result extends to initial conditions that are proper graphs over subdomains of ${\mathbb{R}}^n$.

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\ne 0$. Suppose that the structure vector field $\xi$ of $M$ is an eigen vector field of the Ricci tensor $S$, $S\xi =\beta \xi$, $\beta$ being a function. We study on $M$, a gradient pseudo-Ricci soliton as an extended concept of Ricci soliton, closely related to pseudo-Einstein real hypersurfaces. We show that a $3$-dimensional ruled real hypersurface of $M^2(c),c<0$ admits a non-trivial gradient pseudo-Ricci soliton.

On any closed hyperbolizable 3-manifold, we find a sharp relation between the minimal surface entropy (introduced by Calegari-Marques-Neves) and the average area ratio (introduced by Gromov), and we show that, among metrics g with scalar curvature greater than or equal to -6, the former is maximized by the hyperbolic metric. One corollary is to solve a conjecture of Gromov regarding the average area ratio. Our proofs use Ricci flow with surgery and laminar measures invariant under a PSL(2,R)-action.

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.

We relate the (non)existence of lower scalar curvature bounds to the existence of certain distance-decreasing maps. We also give a sufficient condition for the existence of a limiting scalar curvature measure in the backward limit of a Ricci flow solution.

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\le a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov-Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than $r^{\frac{n+1}{2}}.$ This result not only improves the estimate in [CMZ21b, Theorem 1.3], but also is optimal since the Bryant soliton and Appleton's solitons [Ap17] have exactly this growth rate.

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to e.g.\ geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal $H^1$–error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.

In this short note, we observe that the Bamler-Kleiner proof of uniqueness and stability for 3-dimensional Ricci flow through singularities generalizes to singular Ricci flows in higher dimensions that satisfy an analogous canonical neighborhood property. In particular, this gives a canonical evolution through singularities for manifolds with positive isotropic curvature. The new ingredients we use are the recent classification of higher dimensional $\kappa$-solutions by Brendle, Daskalopoulos, Naff and Sesum, and the maximum principle for the linearized Ricci-DeTurck flow on locally conformally flat manifolds due to Chen and Wu.

We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for $\kappa$-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a quotient of the fundamental group of the regular part of any tangent flow at infinity.

Motivated by Luo's combinatorial Yamabe flow on closed surfaces and Guo's combinatorial Yamabe flow on surfaces with boundary, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.

We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g are constant; 3. g is extremal (not cscK) and one of its generalized scalar curvature is constant.

In this paper, we consider a translating soliton for the mean curvature flow starting from a graph of a function on a domain in a unit sphere which is constant along each leaf of isoparametric foliation. First, we show that such a function is given as a composition of an isoparametric function on the sphere and a function which is given as a solution of a certain ordinary differential equation. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons. Finally, we investigate a domain of the function which is given as a composition of the isoparametric function and the solution of the ordinary differential equation in the case where the number of distinct principal curvatures of the isoparametric hypersurface defined by the regular level set for the isoparametric function is 1, 2, or 3.

In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler [Bam20c]. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in [CCGGIIKLLN10], (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit.

The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian metrics admitting parallel spinors are not necessarily Ricci flat, the parallel spinor flow preserves the vacuum momentum and Hamiltonian constraints and therefore the Einstein and parallel spinor flows coincide on common initial data. Using this result, we provide an initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds, which in turn yields the first initial data characterization of Ricci-flat pp-waves. Furthermore, we explicitly solve the left-invariant parallel spinor flow on simply connected Lie groups, obtaining along the way necessary and sufficient conditions for the flow to be immortal. These are, to the best of our knowledge, the first non-trivial examples of evolution flows of parallel spinors. Finally, we use some of these examples to construct families of $\eta$-Einstein cosymplectic structures and to produce solutions to the left-invariant Ricci flow in three dimensions. This suggests the intriguing possibility of using first-order hyperbolic spinorial flows to construct special solutions of curvature flows in Riemannian signature.

We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.

Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a very large class of extrinsic curvature flows, namely those whose speeds are $\alpha$-homogeneous ($\alpha >0$), elliptic and symmetric with respect to the principal curvatures. We show that these solutions are necessarily convex, and give precise criteria for the speed functions which determine whether these translators are defined on all of ${\mathbb{R}}^n$ or contained in a cylinder. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we are also able to describe the asymptotics of the translator at infinity.

Under broad hypotheses we derive a scalar reduction of the generalized Kähler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819, finishing the classification in complex dimension $2$.

In this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of gradient Ricci solitons that are realized as warped products, from which we know that the base spaces of these products are Ricci-Hessian type manifolds. We study this latter class of manifolds as the most appropriate setting to prove our results.

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the homogeneous Kähler-Ricci flow on rational homogeneous varieties. This description enables us to compute explicitly the maximal existence time for any homogeneous solution and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between numerical invariants associated to ample divisors and numerical invariants arising from solutions of the homogeneous Kähler-Ricci flow. In the particular setting of full flag varieties, we prove that the numerical invariants obtained from solutions of the homogeneous Kähler-Ricci flow can be related to certain well-known invariants which appear in some different contexts, including the global Seshadri constant of ample line bundles, the maximum possible radius of embeddings of symplectic and Kähler balls, and the log canonical threshold of ample $\mathbb{Q}$-divisors. From this, we obtain constraints in terms of the scalar and Ricci curvatures, respectively, for symplectic embeddings of open Euclidean balls, and for the triviality of analytic multiplier ideal sheaves defined by ample $\mathbb{Q}$-divisors.

In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values $g$ are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to $W^{2,2}$ and satisfy $\frac{1}{a}h\le g\le ah$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution. Results for a related non-compact setting are also presented. Various $L^p$ estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature $\ge k$ for $W^{2,2}$ metrics $g$ on closed four manifolds which are bounded in the $L^{\infty }$ sense by $\frac{1}{a}h\le g\le ah$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$.

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with $L^2$ norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the $L^p$ norm for certain $p>2$ of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.

We study solutions to generalized Ricci flow on four-manifolds with a nilpotent, codimension $1$ symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy adapted to this setting, we show that blowdown limits lie in a canonical finite-dimensional family of solutions. The results are new for Ricci flow.

Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle for non-compact spaces. Often it can be avoided if one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead we solve the gauge problem directly in great generality. The techniques and ideas apply to many problems. We use them to solve a well-known open problem in Ricci flow: Strong rigidity of cylinders. Strong rigidity is an illustration of a {\it shrinker principle} that uniqueness radiates out from a compact set. It implies that if one tangent flow at a future singular point is a cylinder, then all tangent flows are. We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism. Strong rigidity relies on gauge fixing and several other new ideas. One of these is "propagation of almost splitting", another is quadratic rigidity in the right gauge, and a third is an optimal polynomial growth bound for PDEs that holds in great generality.

In this paper, we initiate the study of conformal $\eta$-Ricci soliton and almost conformal $\eta$-Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian metric admits conformal $\eta$-Ricci soliton, then the manifold is $\eta$-Einstein and either the soliton vector field $V$ is Killing or it leaves $\phi$ invariant. Here, we have shown the characteristics of the soliton vector field $V$ and scalar curvature when the manifold admitting conformal $\eta$-Ricci soliton and vector field is pointwise collinear with the characteristic vector field $\xi$. Next, we show that a para-Sasakian metric endowed an almost conformal $\eta$-Ricci soliton is $\eta$-Einstein metric if the soliton vector field $V$ is an infnitesimal contact transformation. We have also displayed that the manifold is Einstein if it represents a gradient almost conformal $\eta$-Ricci soliton. We have developed an example to display the alive of conformal $\eta$-Ricci soliton on 3-dimensional para-Sasakian manifold.

Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and often too severely. In a wide range of areas non-compact spaces come with a Gaussian weight and a drift Laplacian. Eigenfunctions are $L^2$ in the weighted space allowing for extremely rapid growth. Rapid growth would be disastrous for many applications. Surprisingly, for very general tensors, manifolds and weights, we show the same polynomial growth bounds that Laplace and Hermite observed for functions on Euclidean space for the standard Gaussian. This covers all shrinkers for Ricci and mean curvature flows. These results open a door for understanding general non-compact spaces. It provides an analytic framework for doing nonlinear PDE on Gaussian spaces where previously the Gaussian weight allowed wild growth that made it impossible to approximate nonlinear by linear. It is key to bound the growth of diffeomorphisms of non-compact manifolds and is the key for solving the "gauge problem". The relative nature of the estimates and the slow growth in the bounds lead to "propagation of almost splitting" that is significantly stronger than pseudo locality and key for applications.

We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.

We show that the solutions to the curvature flow (CF) for curves on the 2-dimensional light cone are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperboles are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant $c$ from being the inner product between its tangent vector field and a fixed vector $v$ of the 3-dimensional Minkowski space. The curve is a soliton solution when $c=0$. We prove that, for each vector $v$ there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Moreover, at each end of such a curve, the curvature is either unbounded or it tends to $0$ or to the constant $c$. Explicitly given soliton solutions are included and some self-similar solutions on the light cone, are visualized.

We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\ell }$, the round $(n+\ell )$-sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A{|}^2<\frac{1}{n-2}|H{|}^2+4K$ when $n\ge 8$, $|A{|}^2<\frac{4}{3n}|H{|}^2+\frac{n}{2}K$ when $n=7$, and $|A{|}^2<\frac{3(n+1)}{2n(n+2)}|H{|}^2+\frac{2n(n-1)}{3(n+1)}K$ when $n=5$ or $6$. This condition is related to a theorem of Li and Li [Arch. Math., 58:582–594, 1992] which states that the only $n$-dimensional minimal submanifolds of $S_K^{n+\ell }$ satisfying $|A{|}^2<\frac{2n}{3}K$ are the totally geodesic $n$-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of $S_K^{n+1}$ satisfying the pinching condition is diffeomorphic either to the sphere $S^n$ or to the connected sum of a finite number of handles $S^1\times S^{n-1}$. The results are sharp when $n\ge 8$ due to hypersurface counterexamples.

Consider a Riemannian manifold $(M^m,g)$ whose volume is the same as the standard sphere $(S^m,g_{round})$. If $p>\frac{m}{2}$ and ${\int }_M{\left\{Rc-(m-1)g\right\}}_{-}^{p}dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^m,g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.

In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola ${\mathcal{H}}^1(1)$, of center at origin and radius $1$, in the $2$ dimensional Lorentz-Minkowski plane ${\mathbb{R}}_1^2$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of ${\mathcal{H}}^1(1)$, as time tends to infinity.

The main focus of the paper is the investigation of moduli space of left invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the left invariant metrics allows us to use the algebraic approach. However, the geometrical tools, such as classification of hyperbolic plane conics, will often be required. For metrics that we obtain in the classification, we investigate geometrical properties: curvature, Ricci tensor, sectional curvature, holonomy and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as classification of pseudo-Kahler and ppwave metrics. We get the description of parallel symmetric tensors for each metric and showthat they are derived from parallel vector fields. Finally, we investigate the totally geodesic subalgebras by showing that for any subalgebra of the observed algebra there exists a metric that makes it totally geodesic.

The Type IIB flow is a flow of conformally balanced complex manifolds introduced by Phong, Picard, and Zhang, about whose singularities little is as yet known. We formulate convergence criteria for the Gromov-Cheeger-Hamilton convergence of sequences of Hermitian manifolds, and apply them to precompactness theorems and the existence of singularity models for the Type IIB flow, in analogy with Hamilton's classic compactness theorems and classification of singularities for the Ricci flow.

In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a constant weighted mean curvature hypersurface in the Gaussian shrinking Ricci soliton (not necessarily properly immersed) with a certain assumption on the integral of the second fundamental form must be a generalized cylinder.

In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal $G$-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.

We prove that smooth convex $\alpha$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\le CH(x,t)(H(x,t)|x-y|+1{)}^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.

In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure ${\tilde{C}}_{G,\Psi ,\lambda }(\Omega ,\cdot )$ of convex body $\Omega$ in ${\mathbb{R}}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $\eta$-Ricci soliton. Here, we have investigated the nature of the conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $\eta$-Einstein Kenmotsu manifold admitting conformal $\eta$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $\eta$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $\eta$-Ricci soliton on Kenmotu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal $\eta$-Ricci soliton on Kenmotsu manifold.

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3\times \mathbb{R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3\times \mathbb{R}$, $g_{\lambda }^{+}$ is a shrinking while $g_{\lambda }^{+}$ and $g_{\mu }$ are expanding and also $g_0^1,g_0^2,g_0^3$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_4$ only $g_1^{\lambda },g_2^{\lambda }$ have Lorentz Ricci solitons and $g_2^{\lambda }$ is a shrinking.

It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n+1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting $\ast$-Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold $M$ admits $\ast$-Ricci-Yamabe soliton $(g,V,\lambda ,\alpha ,\beta )$ under certain conditions on the soliton vector field $V$, then $M$ is $\ast$-Ricci flat, positive Sasakian and the transverse geometry of $M$ is Fano. In addition, the Sasakian 3-metric $g$ is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field $V$ is an infinitesimal automorphism of the contact metric structure.

Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {\em almost nonpositive $k$-Ricci curvature}, which is weaker than the existence of a Kähler metric with nonpositive $k$-Ricci curvature. When $k=1$, this is just the {\em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with $k$-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact Kähler manifold of almost nonpositive $k$-Ricci curvature must have nef canonical line bundle.

In 1994 Velazquez constructed a smooth $O(4)\times O(4)$ invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velazquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short time existence of Velazquez formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velazquez-Stolarski we therefore show that there exists a solution $\{M_t^7\subset {\mathbb{R}}^8\mid -t_0<t<t_0\}$ that has an isolated singularity at the origin $0\in {\mathbb{R}}^8$, and at $t=0$; moreover, the mean curvature is uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity.

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane ${\mathcal{H}}^n(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space ${\mathbb{R}}_1^{n+1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of ${\mathcal{H}}^n(1)$ as time tends to infinity.

These lecture notes aim to present some of the ideas behind the recent (conditional) existence and (weak-strong) uniqueness theory for mean curvature flow. Focusing on the simplest case of the evolution of a single closed hypersurface allows for a self-contained and concise presentation, which is accessible for beginning graduate students with some background in PDEs and only requires basic measure theory.

In this article we study the breathers of the mean curvature flow in the Euclidean space. A breather is a solution to the mean curvature flow which repeats itself up to isometry and scaling once in a while. We prove several no breather theorems in the noncompact category, that is, under certain conditions, a breather of the mean curvature flow must be a solitonic solution (self-shrinker, self-expander, or translator).

Given an $\infty$-harmonic function $u_{\infty }$ on a domain $\Omega \subseteq {\mathbb{R}}^2$, consider the function $w=-\log |\nabla u_{\infty }|$. If $u_{\infty }\in C^2(\Omega )$ with $\nabla u_{\infty }\ne 0$ and $\nabla |\nabla u_{\infty }|\ne 0$, then it is easy to check that (1) the streamlines of $u_{\infty }$ are the level sets of $w$ and (2) $w$ solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, $w$ is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of $u_{\infty }$ by $p$-harmonic functions, the use of conjugate $p^{\prime }$-harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of $|\nabla u_{\infty }|$ arises as a by-product.

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset.

In this short note we obtain some local and global upper bounds for the Hessian of a positive solution to the conjugate heat equation coupled with the Ricci flow.

The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.

In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to $S^{n-1}\times \mathbb{R}$ in the pointed-Gromov-Hausdorff topology must itself be isometric to $S^{n-1}\times \mathbb{R}$.

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler's precompactness theorem, as well as every singular Ricci flow from Kleiner-Lott, is a weak solution in the sense of Haslhofer-Naber. We also generalize all path-space estimates from Haslhofer-Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.

We show that $S^2\times S^2$ is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for $S^2\times N$, where $N$ belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.

We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478v2] for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions.

Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

In this paper, first we define Clairaut Riemannian map between Riemannian manifolds by using a geodesic curve on the base space and find necessary and sufficient conditions for a Riemannian map to be Clairaut with a non-trivial example. We also obtain necessary and sufficient condition for a Clairaut Riemannian map to be harmonic. Thereafter, we study Clairaut Riemannian map from Riemannian manifold to Ricci soliton with a non-trivial example. We obtain scalar curvatures of $range{F}_{\ast }$ and $(range{F}_{\ast }{)}^{\perp }$ by using Ricci soliton. Further, we obtain necessary conditions for the leaves of $range{F}_{\ast }$ to be almost Ricci soliton and Einstein. We also obtain necessary condition for the vector field $\dot{\beta }$ to be conformal on $range{F}_{\ast }$ and necessary and sufficient condition for the vector field $\dot{\beta }$ to be Killing on $(range{F}_{\ast }{)}^{\perp }$, where $\beta$ is a geodesic curve on the base space of Clairaut Riemannian map. Also, we obtain necessary condition for the mean curvature vector field of $range{F}_{\ast }$ to be constant. Finally, we introduce Clairaut anti-invariant Riemannian map from Riemannian manifold to Kähler manifold, and obtain necessary and sufficient condition for an anti-invariant Riemannian map to be Clairaut with a non-trivial example. Further, we find necessary condition for $range{F}_{\ast }$ to be minimal and totally geodesic. We also obtain necessary and sufficient condition for Clairaut anti-invariant Riemannian maps to be harmonic.

We establish a pointwise estimate of A along the mean curvature flow in terms of the initial geometry and the jHAj bound. As corollaries we obtain the extension theorem of HA and the blowup rate estimate of HA.

We construct several examples related to the scaling limits of energy minimizers and gradient flows of surface energy functionals in heterogeneous media. These include both sharp and diffuse interface models. The focus is on two separate but related issues, the regularity of effective surface tensions and the occurrence of zero mobility in the associated gradient flows. On regularity we build on the theory of Goldman, Chambolle and Novaga to show that gradient discontinuities in the surface tension are generic for sharp interface models. In the diffuse interface case we only show that the laminations by plane-like solutions satisfying the strong Birkhoff property generically are not foliations and do have gaps. On mobility we construct examples in both the sharp and diffuse interface case where the homogenization scaling limit of the $L^2$ gradient flow is trivial, i.e. there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed by Novaga and Valdinoci for forced mean curvature flow.

We formulate and solve the existence problem for Ricci flow on a Riemann surface with initial data given by a Radon measure as volume measure. The theory leads us to a large class of new examples of nongradient expanding Ricci solitons, including the first example of a nongradient Kaehler Ricci soliton. It also settles the question of whether a smooth flow for positive time that attains smooth initial data in a distance metric sense must be smooth down to the initial time. We disprove this by giving an example of a complete Ricci flow starting with the Euclidean plane that is not the static solution.

Perelman has proved that there cannot exist a nontrivial breather for Ricci flow on a closed manifold. Here we construct nontrivial expanding breathers for Ricci flow in all dimensions when the underlying manifold is allowed to be noncompact.

Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established by Gu et al., we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.

We establish a uniqueness result for the $[\phi ,{\vec{e}}_3]$-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk proved that, if $\Sigma$ is a properly embedded translating soliton with locally bounded genus, and ${\mathcal{C}}^{\infty }$-asymptotic to two vertical planes outside a cylinder, then $\Sigma$ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of $[\phi ,{\vec{e}}_3]$-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.

By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac{d{a}^2}{h(a^2)}+a^2{g}_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in ${\mathbb{R}}^n$ for any $n\ge 2$. More precisely for any $\lambda \ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions $h\in C^2((0,\infty );{\mathbb{R}}^{+})$ for the equation $2r^{2}h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+r{h}_r(r)(r{h}_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty )$ satisfying $\underset{\begin{array}{c}r\to 0\end{array}}{\lim }{r}^{\sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.

We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the $L^2$-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.

We give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian metrics on complex manifolds. We emphasize open problems and new directions.

Let $(M,J_0)$ be a Fano manifold which admits a Kähler-Ricci soliton, we analyze the behavior of the Kähler-Ricci flow near this soliton as we deform the complex structure $J_0$. First, we will establish an inequality of Lojasiewicz's type for Perelman's entropy along the Kähler-Ricci flow. Then we prove the convergence of Kähler-Ricci flow when the complex structure associated to the initial value lies in the kernel $Z$ or negative part of the second variation operator of Perelman's entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons in the moduli space of complex structures near $J_0$, and we show that the kernel $Z$ corresponds to the local moduli space of Fano manifolds which are modified $K$-semistable. We also prove an uniqueness theorem for Kähler-Ricci solitons.

In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman's no-local-collapsing theorem.

In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows to find explicitly the translators and describe their main features.

This article grew out of the urge to realize explicit examples of solutions for the Ricci flow as families of isometrically embedded submanifolds, together with its Gromov-Hausdorff collapses. To this aim, we consider the Ricci flow of invariant metrics in a class of flag manifolds. On the one hand, we contrast with a previous result in literature by presenting entire flow lines of invariant metrics realized as orbits of a fixed representation. Indeed, we prove that the subset of realizable metrics has a global attractor with open interior, containing an expressive family of complete flow lines. On the other hand, we prove that certain collapses cannot be realized in any fixed Euclidean space. We provide a detailed picture of the flow, including examples of both realizable and non-realizable flow lines and collapses.

In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in spheres, by K. Choi and C. Mantoulidis, as well as the rigidity of stationary solutions with low Morse index proved by F. Hiesmayr.

Explicit formulae for homogenous Ricci solitons on three-dimensional Lorentzian Bianchi-Cartan-Vranceanu spaces are obtained.

We study complete conformal gradient solitons, a class containing gradient Yamabe solitons and many generalized Yamabe-type structures, including gradient almost Yamabe, gradient k-Yamabe, and gradient h-almost Yamabe solitons, and, after a change of the potential function, gradient Einstein-type manifolds with $\alpha =0$ and $\beta \ne 0$ (in particular, quasi-Yamabe solitons). In this paper, we classify complete nontrivial locally conformally flat conformal gradient solitons. This result contributes to an analogue of Perelman's conjecture for Yamabe-type solitons. Moreover, we show that under nonnegative scalar curvature, every nonflat soliton is rotationally symmetric. We also obtain classifications assuming the Cotton or Cao-Chen tensor vanishes.

This is the second paper in the series to study the generic dynamics of mean curvature flows. We study the initial perturbation of mean curvature flows, whose first singularity is modeled by an asymptotically conical shrinker. The noncompactness of the limiting shrinker creates essential difficulties. We introduce the Feynman-Kac formula to get precise asymptotic behaviour of the linearized rescaled mean curvature equation along an orbit. We also develop the invariant cone method for the noncompact setting for the local dynamics near the shrinker. As a consequence, we prove that after a generic initial perturbation, the perturbed rescaled mean curvature flow avoids the conical singularity.

In this paper we have studied the class of Finsler metrics, called C3-like metrics which satisfy the un-normal and normal Ricci flow equation and proved that such metrics are Einstein.

We consider ancient noncollapsed mean curvature flows in ${\mathbb{R}}^4$ whose tangent flow at $-\infty$ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function $u$ that measures the deviation of the renormalized flow from the round cylinder ${\mathbb{R}}^2\times S^1(\sqrt{2})$ and prove that for $\tau \to -\infty$ we have the fine asymptotics $u(y,\theta ,\tau )=(y^{\top }Qy-2\text{tr}(Q))/|\tau |+o(|\tau {|}^{-1})$, where $Q=Q(\tau )$ is a symmetric $2\times 2$-matrix whose eigenvalues are quantized to be either 0 or $-1/\sqrt{8}$. This naturally breaks up the classification problem for general ancient noncollapsed flows in ${\mathbb{R}}^4$ into three cases depending on the rank of $Q$. In the case $\mathrm{rk}(Q)=0$, generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or $\mathbb{R}\times$2d-bowl. In the case $\mathrm{rk}(Q)=1$, under the additional assumption that the flow either splits off a line or is selfsimilarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be $\mathbb{R}\times$2d-oval or belongs to the one-parameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White, respectively. Finally, in the case $\mathrm{rk}(Q)=2$ we show that the flow is compact and $\mathrm{SO}(2)$-symmetric and for $\tau \to -\infty$ has the same sharp asymptotics as the $\mathrm{O}(2)\times \mathrm{O}(2)$-symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.