Related papers: Mean curvature type flow and sharp Micheal-Simon i…
This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $\beta (\geq 1)$ of homogeneous, either convex or…
We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity.…
We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient…
In this article, we investigate a flow of inverse mean curvature type for capillary hypersurfaces in the half-space. We establish the global existence of solutions for this flow and demonstrate that it converges smoothly to a spherical cap…
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If…
We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to…
We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We prove that if the…
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…
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…
A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact…
In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in $\mathbb{R}^{n+1}$. More precisely, we consider a hypersurface $M$ in $\mathbb{R}^{n+1}$ deformed by a flow along its unit normal with its…
We study mean curvature flow in $\mathbb S_K^{n+1}$, the round sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2} H^{2} + 4 K$ when $n\ge 4$ and $|A|^{2} <…
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the…
In this paper we will discuss how one may be able to use mean curvature flow to tackle some of the central problems in topology in 4-dimensions. We will be concerned with smooth closed 4-manifolds that can be smoothly embedded as a…
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, $k\geq 2$. We establish a codimension…
For a mean curvature flow of complete graphical hypersurfaces $M_{t}=\operatorname{graph} u(\cdot,t)$ defined over domains $\Omega_{t}$, the enveloping cylinder is $\partial\Omega_{t}\times\mathbb{R}$. We prove the smooth convergence of…
In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate…
In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$…
We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed…