Stability & uniqueness

In this paper, we first prove Appleton's conjecuture on the uniqueness of noncollapsed steady solitons constructed in. We also classify non-Ricci-flat $\kappa$-noncollapsed cohomogeneity one steady gradient Ricci solitons in dimension four and five. After passing to a finite cover, in dimension four, they must be the Bryant soliton or Appleton's noncollapsed 4d steady soliton; in dimension five, they must be the Bryant soliton or solitons constructed by Ivey–Dancer–Wang.

In this paper, we study the anisotropic mean curvature flow in the Gaussian space endowed with a Finsler metric obtained from a Minkowski metric by a conformal deformation. We first derive the conformal transformation formulas for the fundamental tensor, the Cartan tensor, and the Chern connection of the conformally related Finsler metrics. Based on these formulas, we express the anisotropic mean curvature in the Gaussian space in terms of the corresponding anisotropic mean curvature with respect to the original Minkowski metric, and derive the associated evolution equation. Finally, we establish the short-time existence and uniqueness of smooth solutions by reformulating the geometric evolution equation as a strictly parabolic quasilinear partial differential equation.

We prove that a complete noncompact gradient shrinking Ricci soliton of dimension at least two is a finite quotient of an Einstein cylinder if its scalar curvature is bounded below by the cylindrical value. The corresponding pointwise strict bound implies compactness. We also establish one-endedness criteria using curvature information at critical points of the potential, including degenerate ones, and obtain sharp one-endedness and rigidity results under global Ricci norm bounds.

We establish uniqueness of Ricci flows with non-negative sectional curvatures and scaling invariant curvature bounds, assuming that the induced distance functions converge to the same initial distance function up to isometry. In the compact case, restricting the size of the initial singular set, we also establish uniqueness under suitable weaker conditions.

We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.

In this paper, we investigate Ricci solitons in the para-Kenmotsu setting with respect to a $\phi$-related metallic semi-Riemannian metric arising from the underlying almost paracontact metric structure, under the assumption that the potential vector field is pointwise collinear with the Reeb vector field. We first establish the relation $\text{tS}=S+2n\beta (g-\eta \otimes \eta )$ between the Ricci tensors of the original metric $g$ and the $\phi$-related metallic metric $\mathrm{tg}$. We then show that the Ricci soliton equation implies that the original metric $g$ is of $\eta$-Einstein type. By combining this condition with the contracted Bianchi identity, we obtain a rigidity result implying $f=0$, $\lambda =\frac{2n}{C}$, $S=-2ng$, and consequently $\text{tS}=-\frac{2n}{C}\mathrm{tg}$. Thus, every Ricci soliton in this setting whose potential vector field is pointwise collinear with the Reeb vector field is necessarily trivial, and both $g$ and $\mathrm{tg}$ are Einstein. Finally, we provide two five-dimensional para-Kenmotsu examples: one realizing the resulting trivial Einstein Ricci soliton and another showing that failure of the required Einstein condition prevents the existence of such a soliton.

We prove that, in every dimension, the focal set of the potential function of a complete gradient shrinking Ricci soliton with constant scalar curvature is totally geodesic whenever it has dimension two or three, and that every complete five-dimensional gradient shrinking Ricci soliton with constant scalar curvature is rigid. The first result follows from a Bochner identity for the covariant derivative of the Ricci tensor at the focal set. For the second, when the scalar curvature is twice the soliton constant, we prove that total geodesy implies rigidity through an integral identity on the level hypersurfaces; the case of scalar curvature three times the soliton constant is treated by a separate argument. As further applications, we combine our methods with the rigidity theorem in the recent preprint of Li, Li, Ou, Qu, Wu and Zhou to obtain rigidity in arbitrary dimension when the scalar curvature is twice or three times the soliton constant, and under integral Weyl pinching for four-dimensional focal sets.

We study curvature flows for the dual Orlicz–Minkowski problem for strictly convex capillary hypersurfaces in the Euclidean half-space. The prescribed function has the form $\phi (h)G(X)$, where $X=\nabla h+h\nu$ and $G$ may be nonradial and nonhomogeneous. For acute contact angles, the evenness assumption together with either of the two integral conditions yields a smooth weighted-volume-preserving flow for all $t>0$ and a smooth solution to the dual Orlicz–Minkowski equation. Without the evenness assumption, two strict spherical-cap barriers yield the same conclusions for an unnormalized flow, whose limiting hypersurfaces solve the dual Orlicz–Minkowski equation with $c=1$. A uniform strict monotonicity condition under rescaling ensures uniqueness for $c=1$ and convergence of the unnormalized flow in any $C^k$ norm at an exponential rate. The problem considered here includes the capillary $L_p$-Minkowski and $L_p$ dual Minkowski problems as special cases.

We study global smooth Calabi flows on polarized Kähler surfaces $(M,L)$, assuming uniformly bounded Sobolev constants and the algebraic conditions $\chi ({\mathcal{O}}_M)=1$ and $c_1(M)\cdot c_1(L)>0$. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and $|S|\le 1$. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.

We prove strong uniqueness of tangent flows at the first singular time of a closed Ricci flow when one tangent flow is generated by a complete smooth asymptotically conical Ricci shrinker. The modified flow converges to this shrinker in a fixed gauge, with quantitative estimates on expanding domains. The main ingredients are a Łojasiewicz-Simon inequality for Perelman's entropy, based on the global weighted divergence-free gauge constructed in the preceding paper, and estimates controlling the geometric radius under Ricci flow.

We develop the weighted Sobolev and Hölder analysis needed to study perturbations of complete asymptotically conical (AC) Ricci shrinkers. Weighted elliptic estimates yield a global weighted divergence-free slice in spaces that retain the asymptotic data at infinity. As a geometric application, we prove pointed Gromov-Hausdorff rigidity of nontrivial AC shrinkers with vanishing infinitesimal solitonic deformation space.

In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local $\mu$-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the $G_2$-Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

Let $(M^n,g,f)$ be a complete connected shrinking gradient Ricci soliton with constant scalar curvature, satisfying $$\text{Ric}+{\nabla }^{2}f=\frac{1}{2}g.$$ We prove that if the focal set of $f$ is totally geodesic, then $M$ is rigid. Equivalently, its universal cover is a product of a compact Einstein manifold and a Euclidean space.

We study ancient solutions to curve shortening flow on a class of warped product surfaces. For closed embedded ancient solutions whose curves are not null-homotopic, we prove that graphicality at every time is equivalent to a uniform length bound as $t\to -\infty$, and that these properties hold whenever the length is $o(\sqrt{-t})$. Under an additional positivity condition on the linearized height equation, every such solution with bounded length is a family of slices. We derive backward convergence to a limiting circle from the length bound and prove slice rigidity by an $L^2$ estimate for the height perturbation.

We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an $\epsilon$-uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the $L^{\infty }$ norm.

We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ${\mathbb{H}}^{n+1},n\ge 2$, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the $C^{1,\alpha }\cap W^{2,k+1}$ approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For $0\le k\le n-1$, we construct weak flows preserving the $k$-th modified quermassintegral ${\tilde{W}}_k$ with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of ${\tilde{W}}_{k+1}$ for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.

In this paper, we prove a uniform pseudolocality theorem for smooth complete Ricci flows whose initial metrics have Ricci curvature bounded below and uniformly positive injectivity radius. No a priori curvature bound is assumed on the flow. As a consequence, we obtain strong uniqueness for smooth complete Ricci flows from a complete metric with the above geometric bounds. We also prove that every smooth complete Ricci flow from a complete Ricci-flat metric with uniformly positive injectivity radius is stationary. The same conclusion holds for any complete flat initial metric. In particular, we give an affirmative answer to Chen's question on the strong uniqueness of Ricci flow from Euclidean space in every dimension.

For $1\le r\le n-1$ we study shrinking self-similar ends of the $r$-mean curvature flow, $S_r=c⟨X,N⟩$ with $c<0$, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect $x{u}^{\prime }-u$ is strictly negative, which yields $r$-admissibility, and the profile has a complete asymptotic expansion for $u(x)/x$ in powers of $x^{-(r+1)}$, valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly $r$-admissible cones whose links have constant principal curvatures.

We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial $Q$-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao–Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.