Related papers: Closed mean curvature flows with asymptotically co…

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…

In this paper, we study the mean curvature type flow for hypersurfaces in the unit Euclidean ball with capillary boundary, which was introduced by Wang-Xia and Wang-Weng. We show that if the initial hypersurface is strictly convex, then the…

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property…

We develop a min-max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there…

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being…

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In…

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of…

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either…

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…

In [5], Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in $\mathbb{R}^{n+1}$, the entropy is uniquely minimized at the round sphere. They conjectured that, for $2\leq n\leq 6$, the round sphere…

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in $\mathbb{C}\mathbb{P}^m$. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a…

We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a…

We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a…

In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity"…

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…

This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a…

In this article, we study hypersurfaces $\Sigma\subset \mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for…

This paper proves that, in mean curvature flow of a compact surface in a complete $3$-manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by…

In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly…

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and…