Related papers: Sharp pinching estimates for mean curvature flow i…

We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the…

We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on…

We prove almost Strichartz estimates found after adding regularity in the spherical coordinates for Schr\"odinger-like equations. The estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages. Sharpness…

We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower…

We construct an $I$-family of ancient graphical mean curvature flows over a minimal hypersurface in $\mathbb{R}^{n+1}$ of finite total curvature with the Morse index $I$ by establishing exponentially fast convergence in terms of $|x|^2-t$.…

In this paper, we prove that a Riemannian $n$-manifold $M$ with sectional curvature bounded above by $1$ that contains a minimal $2$-sphere of area $4\pi$ which has index at least $n-2$ has constant sectional curvature $1$. The proof uses…

For $n\geq 2$, 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…

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a…

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $\Sigma\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on…

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm…

A priori estimates for the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions are established. As an application, the existence of the corresponding parabolic flow is proved,…

We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gau{\ss}…

In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite…

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity,…

A recent article by Li and Lv considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a…

In this paper, we study the rigidity of $k(\ge 1)$-extremal submanifolds in a sphere and prove various pinching theorems under different curvature conditions, including sectional and Ricci curvatures in pointwise and integral sense.

We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is…

Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is…

In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold…

We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as…