Ricci flow papers from 2023

We prove that a complete solution to the Ricci flow on $M\times [-T,0)$ which has quadratic curvature decay on some end of $M$ and converges locally smoothly to the end of a cone on that neighborhood as $t\nearrow 0$ must be a gradient shrinking soliton.

We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds.

We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in ${\mathbb{R}}^3$, which converges to a plane with multiplicity $2$ as time approaches infinity.

We show that the set of awesome homogeneous metrics on non-compact manifolds is Ricci flow invariant. Moreover, if the universal cover of such awesome homogeneous space is not contractible the Ricci flow has finite extinction time, confirming the Dynamical Alekseevskii Conjecture in this case. We also analyze the long-time limits of awesome homogeneous Ricci flows.

In this article, we recapture the Smale conjecture on a Sasakian $3$-sphere via the Legendrian mean curvature flow. More precisely, we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian $3$-spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in ${\mathbb{S}}^2\times {\mathbb{S}}^3$. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in ${\mathbb{S}}^2\times {\mathbb{S}}^3$. Finally, we will address the rigidity theorem of $2$-dimensional Legendrian self-shrinkers in ${\mathbb{R}}^5$. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in ${\mathbb{C}}^3$ from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.

We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a \textit{dual $Q$ curvature} we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.

We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of ${\mathbb{CP}}^n$, $n\ge 2$ is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive $1$-isotropic curvature.

This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator. We discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

We study the evolution of a Jordan curve on the plane by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Fréchet distance.

We study the behavior of a three-dimensional dynamical system with respect to some set $S$ given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter $a\in (0,1/2)$, as for $S$ it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that $S$ is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and $S$ for all $a\in (0,1/2)$. These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases $a\in \{1/9,1/8,1/6\}$ of generalized Wallach spaces.

In this paper, we explore the positive solutions to the Finslerian nonlinear equation $$\frac{\partial u}{\partial t}={\Delta }^{\nabla u}u+au\log u+bu,$$ which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, using a new comparison theorem developed by the first author, we also establish a local gradient estimate on a non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, as well as finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities and a Liouville-type theorem of such solutions.

Here, we study a level-set forced mean curvature flow with evolving spirals and the homogeneous Neumann boundary condition, which appears in a crystal growth model. Under some appropriate conditions on the forcing term, we prove that the solution is globally Lipschitz. We then study the large time average of the solution and deduce the asymptotic growth rate of the crystal. Some large time behavior results of the solution are obtained.

Giving explicit parametrizations of discrete constant Gaussian curvature surfaces of revolution that are defined from an integrable systems approach, we study Ricci flow for discrete surfaces, and see how discrete surfaces of revolution have a geometric realization for the Ricci flow that approaches the constant Gaussian curvature surfaces we have parametrized.

We consider null mean curvature flow along the standard lightcone in the de Sitter spacetime. This flow was first studied by Roesch–Scheuer along null hypersurfaces for the detection of MOTS, and independently by the author in the specific case of the standard Minkowski lightcone. Similar to the Minkowski case, null mean curvature flow along the de Sitter lightcone can be related to $2d$-Ricci flow for surfaces of genus $0$ by an appropriate rescaling. Building on this rescaling procedure, we analyse singularity formation, asymptotic behavior and ancient solutions to the flow.

We are concerned in this article with a classical topic in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of sectional curvature (resp. holomorphic sectional curvature) of a compact Riemannian manifold (resp. Kähler manifold) can be completely determined by the eigenvalues of its $p$-Laplacian for a single integer $p$? We treat this question under two conditions: gradient shrinking Ricci soliton for Riemannian manifolds and cohomologically Einstein for Kähler manifolds. We show that, with some sporadic unknown cases, this is true for each $p$. Furthermore, we show that the condition of being isospectral can be relaxed to a suitable almost-isospectral version.

We prove that the moduli space of mean convex two-spheres embedded in complete, orientable 3-dimensional Riemannian manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We also study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterisation of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature.

We show that any tangent cone of a singular shrinking Kähler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set in the algebraic sense. Along the way, we give a parabolic proof of Hörmander's $L^2$ estimate, which can be used to solve the $\overline{\partial }$-equation on any singular shrinking Kähler-Ricci soliton.

In this paper, we show that the Calabi flow can be extended as long as the $L^p$ scalar curvature is uniformly bounded for some $p>n$, and on a compact extremal Kähler manifold the Calabi flow with uniformly bounded $L^p(p>n)$ scalar curvature exists for all time and converges exponentially fast to an extremal Kähler metric.

Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scalings. We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.

We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in ${\mathbb{R}}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining our work with results of Brendle and Choi-Haslhofer-Hershkovits-White, we show that any level set flow starting from an embedded surface diffeomorphic to a 2-spheres does not fatten. In fact, we obtain that the problem of evolving embedded 2-spheres via the mean curvature flow equation is well-posed within a natural class of singular solutions. Second, we use our result to remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows starting from any generic embedded surface only incur cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of mean curvature flows that flow through singularities. Among other things, this theory also applies to the innermost and outermost flow of any embedded surface and shows that all singularity models of such flows must have multiplicity one. It also establishes equality of the fattening time with the discrepancy time. Lastly, we obtain a number of further results characterizing a separation phenomenon of mean curvature flows of surfaces.

We prove that for a convex polytope decomposition of a domain in ${\mathbb{R}}^n$, an integral curve of a piecewise analytic vector field is chopped by the decomposition into finitely many pieces. As a consequence, we prove the finiteness of the number of the edge flips in a discrete Yamabe flow.

Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle.

We prove Ilmanen's resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface $M^n\subset {\mathbb{R}}^{n+1}$, $2\le n\le 6$, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.

A variant of the Gauss curvature flow for closed and convex hypersurfaces is considered. We reveal that if the initial hypersurface is pinched enough, then this property is preserved. Furthermore, based on some structure assumptions on the speed function of the shrinking flow, we show that the flow converges to a sphere. This may generalize the result of B. Chow to the possible non-homogeneous curvature flows.

This paper studies singularities of mean curvature flows with integral mean curvature bounds $H\in L^{\infty }{L}_{loc}^p$ for some $p\in (n,\infty ]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$. When $p=\infty$ and $\mathbf{C}$ is a regular cone, we prove that the tangent flow is unique. These results hold for general integral Brakke flows of arbitrary codimension in an open subset $U\subseteq {\mathbb{R}}^N$ with $H\in L^{\infty }{L}_{loc}^p$. For smooth, codimension one mean curvature flows with $H\in L^{\infty }{L}_{loc}^{\infty }$, we also show that, at points where a tangent flow is given by an area-minimizing Simons cone, there is an accompanying limit flow given by a smooth Hardt-Simon minimal surface.

We prove the following result: Let $(M,g_0)$ be a complete noncompact manifold of dimension $n\ge 12$ with isotropic curvature bounded below by a positive constant, with scalar curvature bounded above, and with injectivity radius bounded below. Then there is a finite collection $\mathcal{X}$ of spherical $n$-manifolds and manifolds of the form ${\mathbb{S}}^{n-1}\times \mathbb{R}/G$, where $G$ is a discrete subgroup of the isometry group of the round cylinder ${\mathbb{S}}^{n-1}\times \mathbb{R}$, such that $M$ is diffeomorphic to a (possible infinite) connected sum of members of $\mathcal{X}$. This extends a recent work of Huang. The proof uses Ricci flow with surgery on open orbifolds with isolated singularities.

Motivated by the problem of finding constant scalar curvature Kähler metrics, we investigate a Ricci iteration sequence of Rubinstein that discretizes the pseudo-Calabi flow. While the long time existence of the flow is still an open question, we show that the iteration sequence does exist for all steps, along which the K-energy decreases. We further show that the iteration sequence, modulo automorphisms, converges smoothly to a constant scalar curvature Kähler metric if there is one, thus confirming a conjecture of Rubinstein from 2007 and extending results of Darvas–Rubinstein to arbitrary Kähler classes.

For $n\ge 2$, we construct $I$-dimensional family of embedded ancient solutions to mean curvature flow arise from an unstable minimal hypersurface $\Sigma$ with finite total curvature in ${\mathbb{R}}^{n+1}$, where $I$ is the Morse index of the Jacobi operator on $\Sigma$.

In this article we examine the formation of cylindrical neckpinch singularities in Ricci flow in compact manifolds. Rigorous examples of neckpinch sigularity for rotationally symmetric initial data were first constructed by Angenent and Knopf, and examples with given asymptotic behaviour were constructed by them and Isenberg. Here we discuss the asymptotic behavior of the flow under Type-I assumption for general symmetric initial data, and show the previously constructed asymptotic profiles and are the only possibilities.

We give an algebraic criterion for a nilpotent real Lie algebra and prove that it provides a necessary and sufficient condition for the associated nilpotent Lie group to admit left-invariant Ricci solitons, called nilsolitons. As an application of this result, we generalise Nikolayevsky's criterion for the existence of nilsolitons to nilpotent Lie algebras without nice bases. We further prove a modified version of the Taketomi–Tamaru conjecture for nilpotent Lie groups which gives an obstruction to the existence of nilsolitons.

By carrying out refined curvature estimates, we prove better rigidity theorems of complete noncompact ancient solutions to the mean curvature flow in higher codimension under various Gauss image restriction.

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdière's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

In this paper we study 4d gradient steady Ricci solitons, which are weak $\kappa$-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact $\kappa$-solutions found in [ABDS22]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows.

We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is valid for all times and is insensitive to singularities. The construction is characterized by discretization in time and the approximant, while still exhibiting the possibile formation of finitely many singularities at a finite set of singular times, exists globally and is well behaved and simpler to analyze than a solution of the CSF. A solution of the latter is obtained in the limit. Estimates for a natural (geometric) norm involving length and total absolute curvature allow passage to the limit. Many classical qualitative results about the flow can be recovered by exploiting the simplicity of the approximant and new ones can be proved. The construction also suggests a numerical procedure for the computation of the flow which proves very effective as demonstrated by a series of numerical experiments scattered throughout the paper.

In this survey paper, I discuss some recent progress on the existence and regularity of Brakke flows. These include: an "end-time version" of Brakke's local regularity theorem, which allows to extend the validity of the celebrated regularity theorem by White from limits of smooth mean curvature flows to arbitrary Brakke flows; a global-in-time existence theorem for multi-phase Brakke flows of grain boundaries satisfying suitable $\mathrm{BV}$ regularity in time, in both the unconstrained and the fixed boundary settings, with applications to Plateau's problem for the latter; and the proof that branching singularities of minimal surfaces are a trigger for dynamical instability, in the sense that they may be "perturbed away" by a non-trivial canonical Brakke flow. This note is an extended version of a talk given by the author at the MATRIX Research Institute on the occasion of the workshop entitled "Minimal surfaces and geometric flows: interaction between the local and the nonlocal worlds".

In this paper, we prove that given a hyperbolic polyhedral metric with an inversive distance circle packing, and a target discrete curvature satisfying Gauss-Bonnet formula, there exist a unique inversive distance circle packing which is discrete conformal to the former one. We deform the surface by discrete Ricci flow, and do surgery by edge flipping when the orthogonal circles of some faces are about to be non-compact. The revised weighted Delaunay inequality of hyperbolic case implies the compactness of the orthogonal circle. We use a variational principle of a convex Ricci potential defined on the fiber bundles with cell-decomposition and differential structure based on Teichmüller space to finish the proof.

There has been a conjecture, often attributed to Thurston, which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this paper, for a certain class of almost Fuchsian manifolds, we prove the long-time existence and convergence of the modified mean curvature flow $$\frac{\partial F}{\partial t}=-(H-c)\vec{\nu },$$ which was first introduced by Xiao and the second named author in. As an application, we confirm Thurston's CMC foliation conjecture for such a subclass of almost Fuchsian manifolds.

In this paper we consider $L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a gradient shrinking or steady Kähler-Ricci soliton, then we prove that any pluriharmonic function $u$ on $M$ with $\nabla u\in L^p(M)$ for some $1<p\le 2$ is a constant function.

We consider a locally constrained curvature flow in a static rotationally symmetric space ${\mathbf{N}}^{n+1}$, which was firstly introduced by Hu and Li in the hyperbolic space. We prove that if the initial hypersurface is graphical, then the smooth solution of the flow remains to be graphical, exists for all positive time $t\in [0,\infty )$ and converges to a slice of ${\mathbf{N}}^{n+1}$ exponentially in the smooth topology. Moreover, we prove that the flow preserves static convexity if the initial hypersurface is close to a slice of ${\mathbf{N}}^{n+1}$ in the $C^1$ sense. As applications, we prove a family of weighted geometric inequalities for static convex domains which is close to a slice of ${\mathbf{N}}^{n+1}$ in the $C^1$ sense.

We study the mean curvature flow of smooth $m$-dimensional compact submanifolds with quadratic pinching in the Riemannian manifold $\mathbb{C}{P}^n$. Our main focus is on the case of high codimension, $k\ge 2$. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and moves by translation. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Lastly, we prove a decay estimate that shows that the rescaling converges smoothly to a totally geodesic limit in infinite time. This behaviour is only possible if the dimension of the submanifold is even. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature. This result generalises the work of Pipoli and Sinestrari on the mean curvature flow of submanifolds of the complex projective space.

Special Ricci-Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367–380] involve functions $\tau$ on $M$ and state that, for some function $\alpha$ of the real variable $\tau$, the sum of $\alpha \nabla d\tau$ and the Ricci tensor equals a functional multiple of the metric $g$, while $\alpha \nabla d\tau$ itself is assumed to be nonzero almost everywhere. Three well-known obvious "standard" cases are provided by (non-Einstein) gradient Kähler-Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler-Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $\alpha =2\cot \tau$, or $\alpha =2\coth \tau$, or $\alpha =2\tanh \tau$. We also use the Cartan-Kähler theorem to prove that these three types are actually realized in a "nonstandard" way.

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on $r_M(x)$, the smallest eigenvalue of the Ricci tensor ${\text{ric}}_x$ in $x$}, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat $3$-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

We prove that for any discrete curvature satisfying Gauss-Bonnet formula, there exist a unique up to scaling inversive distance circle packing in the discrete conformal equivalent class, whose polyhedral metric meets the target curvature. We prove it by constructing diffeomorphism between fiber bundles with cell decomposition based on Teichmüller spaces, and each discrete conformal equivalent class is a fiber passing through finite cell with respect to triangulations, which means we can do surgery on the discrete Ricci flow by edge flipping using a generalized Ptolemy equation to ensure it converge and never blow up.

We study the subsequential convergence of singular solutions to the Ricci flow with prescribed constant in space geodesic curvature on compact surfaces with boundary. Furthermore, we show that in the particular case of rotational symmetry, this convergence does not depend on the sign of the geodesic curvature of the boundary.

We develop a general theory of flows in the space of Riemannian metrics induced by neural network gradient descent. This is motivated in part by recent advances in approximating Calabi-Yau metrics with neural networks and is enabled by recent advances in understanding flows in the space of neural networks. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel, a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman's formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi-Yau metrics, as is expected since the associated neural networks do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of neural network metric flows therefore explains why neural networks are better at learning Calabi-Yau metrics than fixed kernel methods, such as the Ricci flow.

We establish a structure result for the isometry group of non-compact, homogeneous manifolds admitting an immortal homogeneous Ricci flow solution.

The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In this paper we construct two distinct examples of singularities along the flow. First, we find a finite time singularity, ruling out long time existence of the flow in general. Next we show long time existence of the flow with a Calabi symmetry assumption on the blowup of ${\mathbb{P}}^n$, $n\ge 3$, if one assumes supercritical phase. Using this, we find an example where a singularity occurs at infinite time along the destabilizing subvariety in the semi-stable case.

We define the hyperbolic Yamabe flow and obtain some properties of its stationary solutions, namely, of hyperbolic Yamabe solitons. We consider immersed submanifolds as hyperbolic Yamabe solitons and prove that, under certain assumptions, a hyperbolic Yamabe soliton hypersurface is a pseudosymmetric or a metallic shaped hypersurface. We characterize the hyperbolic Yamabe soliton factor manifolds of a multiply twisted, multiply warped, doubly warped, and warped product manifold and provide a classification for a complete gradient hyperbolic Yamabe soliton factor manifold. We also determine the conditions for the factor manifolds to be hyperbolic Yamabe solitons if the manifold is a hyperbolic Yamabe soliton and illustrate this result for a physical model of the universe, namely, for the Robertson–Walker spacetime.

In this paper, we consider the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. We obtain the monotonicity of parabolic frequency for the solution of two nonlinear parabolic equations with bounded Ricci curvature, then we apply the parabolic frequency monotonicity to get some integral type Harnack inequalities and we use -K1 instead of the lower bound 0 of Ricci curvature from Theorem 4.3 in 16, where K1 is any positive constant.

We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [BCDMZ21] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [ZZ23] under different assumptions.

The metric flow is introduced and extensively studied by Bamler [Bam20b, Bam20c], especially as an $\mathbb{F}$-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the $\mathbb{F}$-convergence of a sequence of $\mathbb{F}$-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in a forthcoming work of the authors [CMZ23].

We prove an abstract structure theorem for weighted manifolds supporting a weighted $f$-Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions.

In the paper, we analysis the asymptotic behavior of noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M,g)$ with nonnegative curvature operator away from a compact set $K$ of $M$. In particular, we prove: any $4d$ noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^4,g)$ with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of $(M^4,g)$, which converges subsequently to a family of shrinking quotient cylinders.

All known examples of simply-connected gradient Kähler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\text{SS}$. In this article, we consider the case that $f$ and $\text{SS}$ are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.

This paper is devoted to the investigation of the monotonicity of parabolic frequency functional under conformal Ricci flow defined on a closed Riemannian manifold of constant scalar curvature and dimension not less than 3. Parabolic frequency functional for solutions of certain linear heat equation coupled with conformal pressure is defined and its monotonicity under the conformal Ricci flow is proved by applying Bakry-Emery Ricci curvature bounds. Some consequences of the monotonicity are also presented.

In this paper, we obtain the asymptotic expansion for the analogue of the bowl-soliton for a large `nondegenerate' class of fully nonlinear curvature flows. We use this to show the uniqueness of these bowl-type solitons in their asymptotic class. We also give examples to illustrate the situation for `degenerate' speeds and how they different they can be. Finally, we show how to construct `wing-like' solitons for these flows, which are complete, connected translators that are not graphical, entire or convex. We also obtain asymptotic expansions for them to show the variety of solutions that one can obtain depending on the choice of speed function.

We address the classification of ancient solutions to fully nonlinear curvature flows for hypersurfaces. Under natural conditions on the speed of motion we classify ancient solutions which are convex, noncollapsing, uniformly two-convex and noncompact. There are exactly two possibilities – every such solution is either a self-similarly shrinking cylinder, or else is a rotationally symmetric translating soliton. For a large class of flows this yields a complete classification of the blow-up limits that can arise at a singularity of a solution which is compact, embedded and two-convex.

We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding-Minicozzi respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds.

We establish geometric regularity for Type I blow-up limits of the Kähler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-$W_1$ distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than $4$.

We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano Kähler-Ricci flow to general finite time solutions of the Kähler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the Kähler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kähler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in for general finite time solutions of the Kähler-Ricci flow.

We prove a conjecture by Hoffman, White, and the first author regarding the uniqueness of pitchfork and helicoid translators of the mean curvature flow in ${\mathbb{R}}^3$. We employ an arc-counting argument motivated by Morse-Radó theory for translators and a rotational maximum principle. Applications to the classification of semigraphical translators in ${\mathbb{R}}^3$ and their limits are discussed, strengthening compactness results of the first author with Hoffman-White and with Gama-Moller.

We give a new proof of the Smale conjecture for ${\mathbb{RP}}^3$ and all lens spaces using minimal surfaces and min-max theory. For ${\mathbb{RP}}^3$, the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

In the present article, we characterize Clairaut conformal submersions whose total manifolds admit a Ricci soliton and provide a non-trivial example of such Clairaut conformal submersions. We firstly calculate scalar curvature and Ricci tensors of total manifolds of Clairaut conformal submersions and provide necessary conditions for the fibres of such Clairaut conformal submersions to be almost Ricci solitons and Einstein. Further, we provide necessary conditions for the base manifold to be Ricci soliton and Einstein. Then, we find a necessary condition for vector field $\dot{\zeta }$ to be conformal vector field and killing vector field. Besides, we indicate that if total manifolds of Clairaut conformal submersions admit a Ricci soliton with the potential mean curvature vector field $H$ of $Ker{\vartheta }_{\ast },$ then the total manifolds of Clairaut conformal submersions admit a gradient Ricci soliton. Finally, by solving Poisson equation, we acquire a necessary and sufficient condition for Clairaut conformal submersions to be harmonic.

In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are $C^{2,\alpha }$-close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.

The vacuum Robinson-Trautman solution admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. We perform a comprehensive classification of solutions exhibiting this property in Einstein's gravity with a massless scalar field, assuming that the solution belongs at least to Petrov-type II and some of the components of Ricci tensor identically vanish. We find that these solutions can be grouped into three distinct classes: (I-a) a natural extension of the Robinson-Trautman family incorporating a scalar hair satisfying the time derivative of the Ricci flow equation, (I-b) a novel non-asymptotically flat solution characterized by two functions satisfying Perelman's pair of the Ricci flow equations, and (II) a dynamical solution possessing $\mathrm{SO}(3)$, $\mathrm{ISO}(2)$ or $\mathrm{SO}(1,2)$ symmetry. We provide a complete list of all explicit solutions falling into Petrov type D for classes (I-a) and (I-b). Moreover, leveraging the massless solution in class (I-a), we derive the neutral Robinson-Trautman solution to the $\mathcal{N}=2$ gauged supergravity with the prepotential $F(X)=-i{X}^0{X}^1$. By flipping the sign of the kinetic term of the scalar field, the Petrov-D class (I-a) solution leads to a time-dependent wormhole with an instantaneous spacetime singularity. Although the general solution is unavailable for class (II), we find a new dynamical solution with spherical symmetry from the AdS-Roberts solution via AdS/Ricci-flat correspondence.

By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an $n$-dimensional Riemannian manifold satisfies $${\int }_{V}R(g)dvo{l}_g\ge 0\text{ and }I(V)\ge I({\mathbb{R}}^n),$$ or $${\int }_{V}R(g)dvo{l}_g\ge 0\text{ and }S(V)\ge S({\mathbb{R}}^n),$$ then $g=g_{{\mathbb{R}}^n}$ on $V$, where $R(g)$ is the scalar curvature of $g$, ${\mathbb{R}}^n$ is Euclidean space, $I(V)$ is the isoperimetric constant of $V$ and $S(V)$ is best constant of $L^2$ logarithmic Sobolev inequality of $V$. Moreover,we also obtain the local ${\mathbb{R}}^n$-rigidity about local Perelman's $\nu$-entropy, and local ${\mathbb{S}}^n$-rigidity (resp. ${\mathbb{H}}^n$-rigidity) theorems regarding the cases concerning $R(g)\ge n(n-1)$ (resp. $R(g)\ge -n(n-1)$), weighted isoperimetric constant and best constant of weighted $L^2$ logarithmic Sobolev inequality for the weighted metric ${\left(\cos \frac{d_g(p,x)}{2}\right)}^{-4}g$ (resp. ${\left(\cosh \frac{d_g(p,x)}{2}\right)}^{-4}g$).

We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the $C^1$ regularity of free boundaries under a uniform thickness assumption.

We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting (Andrews and Bryan, Comm. Anal. Geom., 2011). As in the planar case, the sharp estimate implies control on the curvature, resulting in a direct and efficient proof that simple spherical curves either contract to round points (in finite time) or converge to great circles (in infinite time).

This article studies special solutions to symplectic curvature flow in dimension four. Firstly, we derive a local normal form for static solutions in terms of holomorphic data and use this normal form to show that every complete static solution to symplectic curvature flow in dimension four is Kahler-Einstein. Secondly, we perform an exterior differential systems analysis of the soliton equation for symplectic curvature flow and use the Cartan-Kahler theorem to prove a local existence and generality theorem for solitons.

We proved a Bernstein theorem for ancient solution to symplectic mean curvature flow via the complex phase map .

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.

Let $(M^n,g,f)$ be a Ricci shrinker such that ${\text{Ric}}_f=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi {)}^{-\frac{n}{2}}{e}^{-f}d{v}_g$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $\gamma$ and the measure $\overline{\nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_0$ between the measures $\overline{\nu }$ and $\gamma$, and which also elucidates the rigidity implications resulting from this estimate.

We proved a Bernstein theorem of ancient solutions to mean curvature flow.

This paper continues the investigation of isoperimetric inequalities through volume preserving and area decreasing mean curvature type flows related to conformal Killing vector fields. Results of this kind prior to this paper all studied convex hypersurfaces or hypersurfaces which are starshaped with respect to generalized dilations. This paper is the first to study results of this kind for hypersurfaces which are starshaped with respect to general conformal Killing vector fields perturbed by an isometric Killing vector field and our flows allow us to establish isoperimetric inequalities for a much wider class of hypersurfaces. For example, our results apply to hypersurfaces in ${\mathbb{R}}^{n+1}$ which are far from being starshaped in the traditional sense, but are starshaped with respect to the conformal Killing vector field composed of a dilation and a rotational vector field. The flow considered in this paper is a novel modification of the mean curvature-type flow first introduced by Guan and Li, which was later generalized by Guan-Li-Wang and Li-Pan.

We find Fano threefolds $X$ admitting Kähler-Ricci solitons (KRS) with non-trivial moduli, which are $\mathbb{T}$-varieties of complexity two. More precisely, we show that the weighted K-stability of $(X,{\xi }_0)$ (where ${\xi }_0$ is the soliton candidate) is equivalent to certain GIT-stability. In particular, this provides the first examples of strictly weighted K-semistable Fano varieties. On the other hand, we generalize Koiso's theorem to the log Fano setting. Indeed, we show that the K-stability of a log Fano pair $(V,{\Delta }_V)$ is equivalent to the weighted K-stability of a cone $(Y,{\Delta }_Y,{\xi }_0)$ over it. This also leads to new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups. To achieve these, we establish the weighted Abban-Zhuang estimate generalizing the work of, which gives a lower bound of the weighted stability threshold ${\delta }_{\mathbb{T}}^g(X,\Delta )$. This is an effective way to check the weighted K-semistablity of a log Fano triple $(X,\Delta ,{\xi }_0)$. This estimate is also useful in testing (weighted) K-polystability based on the work of.

The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity assumption for all bounded smooth domains in a complete manifold with its bottom spectrum being suitably large relative to its Ricci curvature lower bound. An immediate implication is the nonexistence of embedded compact minimal hypersurfaces in such manifolds. This nonexistence issue is also considered for steady and expanding Ricci solitons.

We prove a quantitative convergence result of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak-strong uniqueness results for mean curvature flow and convergence of the Allen–Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.

The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. As a byproduct, we obtain an almost constancy for a certain integral quantity concerning scalar curvature at an almost selfsimilar point, which is new even for Ricci flow.

We prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains noncollapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian $2$-disk with a strictly convex boundary. Moreover, we prove that there exists a simple closed geodesic with Morse Index $1$ and $2$. This settles the free boundary analog of Grayson's theorem.

Given an entire $C^2$ function $u$ on ${\mathbb{R}}^n$, we consider the graph of $Du$ as a Lagrangian submanifold of ${\mathbb{R}}^{2n}$, and deform it by the mean curvature flow in ${\mathbb{R}}^{2n}$. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of $(I+(D^{2}u{)}^2{)}^{-1}{D}^{2}u$. Such results were previously known only under the stronger assumption of positivity of $D^{2}u$.

Generalized circle packings were introduced in as a generalization of tangential circle packings in hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow, fractional combinatorial Calabi flow and combinatorial $p$-th Calabi flow for generalized hyperbolic circle packings. We establish several equivalent conditions regarding the longtime behaviors of these flows. This provides effective algorithms for finding the generalized circle packings with prescribed total geodesic curvatures.

In this paper, we produce explicit examples of mean curvature flow of (2m-1)-dimensional submanifolds which converge to (2m-2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in ${\mathbb{C}}^m$ with a $U(m)$-invariant Kähler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them.

We prove that the mean curvature flow of a generic closed embedded hypersurface in ${\mathbb{R}}^4$ or ${\mathbb{R}}^5$ with entropy $\le 2$, or with entropy $\le \lambda ({\mathbb{S}}^1)$ if in ${\mathbb{R}}^6$, encounters only generic singularities.

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor $g:S^1\to {\mathbb{R}}_{+}$, and a given direction and speed, this translator is unique. We then construct an ancient compact solution to anisotropic curve shortening flow, and show that this solution, along with the appropriate translating solution, are the unique solutions to anisotropic curve shortening flow that lie in a slab of a given width and no smaller.

We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider. Finally we explain how recent Ricci flow theory is particularly well adapted to solve these problems.

In this paper, we consider translators (for the mean curvature flow) given by a graph of a function on a symmetric space $G/K$ of compact type which is invariant under a hyperpolar action on $G/K$. First, in the case of $G/K=SO(n+1)/SO(n)$, $SU(n+1)/S(U(1)\times U(n))$, $Sp(n+1)/(Sp(1)\times Sp(n))$ or $F_4/\mathrm{Spin}(9)$, we classify the shapes of translators in $G/K\times \mathbb{R}$ given by the graphs of functions on $G/K$ which are invariant under the isotropy action $K\curvearrowright G/K$. Next, in the case where $G/K$ is of higher rank, we investigate translators in $G/K\times \mathbb{R}$ given by the graphs of functions on $G/K$ which are invariant under a hyperpolar action $H\curvearrowright G/K$ of cohomogeneity two.

We provide a derivative estimate for the pluriclosed flow, controlling higher order derivatives of Chern curvature and torsion using the Chern curvature. Moreover, we derive an estimate for torsion tensor using Chern Ricci curvature in dimension two. And in the Hermitian-symplectic case, we find a monotonic quantity and use it to prove that all Hermitian-symplectic solitons are Kähler Ricci solitons.

In this paper, we show that the singularity type of solutions to the Käher-Ricci flow on a numerically effective manifold does not depend on the initial metric. More precisely if there exists a type III solution to the Kähler-Ricci flow, then any other solution starting from a different initial metric will also be Type III. This generalises previous results by Y. Zhang for the semi-ample case.

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are homothetic self-similar solutions for the inverse mean curvature flow. This classification shows the existence of two classes of non-cylindrical homothetic solitons: one with lightlike rulings and another one with non-lightlike rulings.

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are translating solitons for the inverse mean curvature flow. In particular, we prove the existence of non-cylindrical ruled translating solitons, which contrast with the Euclidean setting.

In this paper, we study the rigidity and ε-regularity theorems of Ricci shrinkers. First we prove the rigidity of the asymptotic volume ratio and local volume around a base point of a non-compact Ricci shrinker. Next we obtain some ε-regularity theorems of local entropy and curvature, which improve the previous corresponding results essentially and use them to study the structure of Ricci shrinkers at infinity. Especially, if the curvature of a non-compact Ricci shrinker satisfies some natural integral conditions, then it is asymptotic to a cone.

We show that for certain one-parameter families of initial conditions in ${\mathbb{R}}^3$, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.

The $L_p$ chord Minkowski problem based on Chord measures and $L_p$ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the $L_p$ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when $p>1$ and $0<p<1$ in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for $0\le p<1$ by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the $L_p$ chord Minkowski problem by replacing $p$ with a fixed continuous function $\phi :(0,\infty )\to (0,\infty )$, and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.

Inversive distance circle packings introduced by Bowers-Stephenson are natural generalizations of Thurston's circle packings on surfaces. To find piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial $p$-th Calabi flow for the Euclidean inversive distance circle packings. Due to the singularities possibly developed by these combinatorial curvature flows, the longtime existence and convergence of these combinatorial curvature flows have been a difficult problem for a long time. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Using the discrete conformal theory recently established by Bobenko-Lutz for decorated piecewise Euclidean metrics on surfaces, we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. This provides effective algorithms for finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.

We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\infty$ to four vertical planes $\{y=\pm b\}$ and $\{y=\pm B\}$, where $0<b\le B<\infty$. We call $b$ and $B$ the inner width and the (outer) width of the translator. We show that for each $b\ge \pi /2$ and each $s>0$, there is a translator in the family with inner width $b$ and with necksize $s$. (We also show that there are no translators with inner width $<\pi /2$ having the properties of the examples we construct.)

We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the singular Ricci flow: given a singular gradient Ricci flow spacetime in the Kleiner-Lott sense, we construct a unique global ambient half-space from it. We also prove the converse, that every global ambient space contains a singular gradient Ricci flow spacetime, thereby completing the correspondence. Our main application is the construction of infinite families of fully non-linear analogues of Perelman's $\mathcal{F}$ and $\mathcal{W}$ functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a "Ricci flow vector field" in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.

We introduce a classification conjecture for $\kappa$-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of ${\mathbb{Z}}_2^2\times {\mathrm{O}}_3$-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-compact gradient shrinking Ricci solitons with superharmonic potential. Moreover, an upper bound for the conjugate radius of Ricci soliton with concircular potential vector field is determined. Finally, it is proved that a non-compact gradient Ricci soliton with a pole and non-negative Ricci curvature is non-shrinking.

We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on ${\mathbb{R}}^3\times S^1.$ More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants.

We develop the asymptotic analysis as $\epsilon \to 0$ for the natural gradient flow of the self-dual $U(1)$-Higgs energies $$E_{\epsilon }(u,\nabla )={\int }_M\left(|\nabla u{|}^2+{\epsilon }^2|F_{\nabla }{|}^2+\frac{(1-|u{|}^2{)}^2}{4{\epsilon }^2}\right)$$ on Hermitian line bundles over closed manifolds $(M^n,g)$ of dimension $n\ge 3$, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows – i.e., integral $(n-2)$-Brakke flows – generalizing results of the last two authors from the stationary case. Given any integral $(n-2)$-cycle ${\Gamma }_0$ in $M$, these results can be used together with the convergence theory developed in previous work of the authors to produce nontrivial integral Brakke flows starting at ${\Gamma }_0$ with additional structure, similar to those produced via Ilmanen's elliptic regularization.

We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic $3$-space ${\mathbb{H}}^3$, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in ${\mathbb{H}}^3$, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in ${\mathbb{H}}^3$ and apply it to prove that properly immersed translators to MCF in ${\mathbb{H}}^3$ are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the $x_3$-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in ${\mathbb{H}}^3$ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in ${\mathbb{H}}^3$ and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in ${\mathbb{R}}^3$.

We make classifications of gradient Ricci solitons $(M,g,f)$ with harmonic Weyl curvature. As a local classification, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein manifold, the Riemannian product of a Ricci flat manifold and an Einstein manifold, a warped product of $\mathbb{R}$ and an Einstein manifold, and a singular warped product of ${\mathbb{R}}^2$ and a Ricci flat manifold. Compared with the previous four-dimensional study in, we have developed a novel method of {\it refined adapted frame fields} and overcome the main difficulty arising from a large number of Riemmannian connection components in dimension$\ge 5$. Next we have obtained a classification of {\it complete} gradient Ricci solitons with harmonic Weyl curvature. For the proof, using the real analytic nature of $g$ and $f$, we elaborate geometric arguments to fit together local regions.

We study mean convex mean curvature flow $M_s$ of local spacelike graphs in the flat slicing of de Sitter space. We show that if the initial slice is of non-negative time and is graphical over a large enough ball, and if $M_s$ is of bounded mean curvature, then as $s$ goes to infinity, $M_s$ becomes graphical in expanding balls, over which the gradient function converges to $1$. In particular, if $p_s$ is the point lying over the center of the domain ball in $M_s$, then $(M_s,p_s)$ converges smoothly to the flat slicing of de Sitter space. This has some relation to the mean curvature flow approach to the cosmic no hair conjecture.

We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler–Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., ${\rho }_{\omega }^k=\lambda \Omega$. In particular, when $k=1$, under some condition on the maximal domain of definition of canonical coordinates, we show that $\lambda$ is forced to be positive. Moreover, for arbitrary $k$, we prove two additional results. Namely, if $\omega$ and $\Omega$ are induced by a flat metric, then $\omega$ is Ricci-flat. Finally, if a Kähler-Ricci soliton $\Omega$ arises as Kähler–Ricci iteration of a metric $\omega$ induced by a complex space form, then the Kähler–Ricci soliton is forced to be trivial, that is, Kähler–Einstein. These three theorems extend well known results on Kähler–Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.

We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Kähler-Ricci flow. As an application, we obtain a monotonicity formula.

In order to reconcile the wave-function collapse in quantum mechanics with the finiteness of signals' propagation in general relativity, we delve into a stochastic version of the Ricci flow and study its non-relativistic limit in presence of matter. We hence derive the Diósi-Penrose collapse model for the wave-function of a quantum gas. The procedure entails additional parameters with respect to phenomenological models hitherto accounted for, including the temperature of the gas and the cosmological constant, in turn related to the stochastic gravitational noise responsible for the collapse.

After an introductory chapter on the quantum supersymmetric string, in which particular attention will be devoted to the techniques via which phenomenologically viable models can be obtained from the ultraviolet microscopic degrees of freedom, and a brief review of the swampland program, the technical tools required to deal with geometric flows will be outlined. The evolution of a broad family of scalar and metric bubble solutions under Perelman's combined flow will be then discussed, together with their asymptotic behaviour. Thereafter, the geometric flow equations associated to a generalised version of Perelman's entropy function will be derived and employed in defining the action-induced flow associated to a given theory for a scalar field and a dynamical metric. The problem of preserving Einstein field equations along the corresponding moduli space trajectories will be cured by allowing a supplementary energy-momentum tensor term to appear along the flow. In a particular example, such contribution will be shown to precisely reproduce the infinite tower of states with exponentially dropping masses postulated by the distance conjecture.

Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3,g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on ${\mathbb{R}}^3$.

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{SU}(3)/{\mathrm{T}}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the tetrahedron $\mathcal{T}:=x+y+z=1$, in this paper, we describe all regions $\mathcal{R}\subset \mathcal{T}$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion's. We study the dynamics of such regions under the projected Ricci flow in the plane $(x,y)$, concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $\mathrm{SU}(3)/{\mathrm{T}}^2$, further studying its evolution under the Ricci flow on the base.

We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure that the renormalized volume and the ADM mass surface integral are well-defined separately. We prove several positivity results for the volume-renormalized mass. We also use it to define a renormalized Einstein–Hilbert action and a renormalized expander entropy which is nondecreasing under the Ricci flow. Further, we show that local maximizers of the entropy are local minimizers of the volume-renormalized mass.

It is shown that on compact $Spin(7)$–manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$–torsion connection $R\in S^2{\Lambda }^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$–torsion connections simultaneously. Precise conditions for a compact $Spin(7)$–manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$–torsion connection. It is shown that a compact $Spin(7)$–manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$–manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$–structure.

Curvature properties of the characteristic connection on an integrable $G_2$ manifold are investigated. We consider integrable $G_2$ manifold of constant type, i.e. the scalar product of the exterior derivative of the $G_2$ form with its Hodge dual is a constant. We show that on an integrable $G_2$ manifold of constant type with $G_2$-instanton characteristic curvature and vanishing Ricci tensor the torsion 3-form is harmonic. Consequently, we prove that the characteristic curvature is symmetric in exchange the first and the second pair and Ricci flat if and only if the three-form torsion is parallel with respect to the Levi-Civita and to the characteristic connection simultaneously and this is equivalent to the condition that the characteristic curvature satisfies the Riemannian first Bianchi identity. We find that the Hull connection is a $G_2$-instanton exactly when the torsion is closed. We observe that any compact integrable $G_2$ manifold with closed torsion is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the characteristic connection. In particular, this vector field is an infinitesimal automorphism of the $G_2$ structure and preserves the torsion three form.

The second author and H. Yin have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a nonatomic Radon measure as a volume measure. This led to the discovery of a large array of new expanding Ricci solitons. In this paper we use the recent uniqueness theory in this context, also developed by the second author and H. Yin, to give a complete classification of all expanding Ricci solitons on surfaces. Along the way, we prove a converse to the existence theory that is not constrained to solitons: every complete Ricci flow on a surface over a time interval $(0,\epsilon )$ admits a $t\downarrow 0$ limit within the class of admissible initial data. This makes surfaces the first nontrivial setting for Ricci flow in which a bijection can be given between the entire set of complete Ricci flows over maximal time intervals $(0,T)$, and a class of initial data that induces them.

In this paper we study conformal solitons for the mean curvature flow in hyperbolic space ${\mathbb{H}}^{n+1}$. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field $-{\partial }_0$. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability, and address the case of noncompact boundaries contained between two parallel hyperplanes of ${\partial }_{\infty }{\mathbb{H}}^{n+1}$. We conclude by proving rigidity results for bowl and grim-reaper cylinders.

It is observed that on a compact almost complex Calabi-Yau with torsion 6-manifold the Nijenhuis tensor is parallel with respect to the torsion connection. If the torsion is closed then the space is a compact generalized gradient Ricci soliton. In this case, the torsion connection is Ricci-flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. On a compact almost complex Calabi-Yau with torsion 6-manifold it is shown that the curvature of the torsion connection is symmetric on exchange of the first and the second pairs and has vanishing Ricci tensor if and only if it satisfies the Riemannian first Bianchi identity.

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in S^2{\Lambda }^2$ then the scalar curvature of a $\nabla$-Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity $R(X,Y,Z,V)=R(Z,Y,X,V)$ must be flat.

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci imply the $C^{1+\alpha }$-regularity of the flow and then provide a regularity argument which improves this into $C^{2+\alpha }$-regularity of the flow. The regularity step from $C^{1+\alpha }$ into $C^{2+\alpha }$ does not rely on convexity and can probably be adopted to more general setting.

Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral iteration argument which utilizes Hamilton's gradient estimate, heat kernel Gaussian bounds and parabolic Harnack inequality. In this paper, we show that the sharp Li-Yau estimate can actually be obtained directly following the classical maximum principle argument, which simplifies the proof in [26]. In addition, we apply the same idea to the heat and conjugate heat equations under the Ricci flow and prove some Li-Yau type estimates with optimal coefficients.

Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the ${\pi }_2$-systole of $M$ (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most $8\pi$. We prove the following gap theorem: if $M$ is further not a quotient of $S^2\times S^1$, then the ${\pi }_2$-systole of $M$ is no greater than an improved constant $c\approx 5.44\pi$. This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow.

The Bach tensor is classically defined in dimension 4, and work from J. Bergman and others shows that $B=\frac{1}{2}U+\frac{1}{6}V$ where $U$ and $V$ are more basic 2-tensors, which are symmetric, divergence-free, algebraically independent, and quadratic in the Riemann tensor. In this paper, we extend H.-D. Cao and Q. Chen's results for Bach-flat gradient shrinking Ricci solitons to solitons with $\mathfrak{B}=\alpha U+\beta V=0$.