Related papers: Biharmonic hypersurfaces with bounded mean curvatu…

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the…

For any $H$ in (0,1/2), we construct complete, non-proper, stable, simply-connected surfaces embedded in $H^2xR$ with constant mean curvature $H$.

For each $k\geq2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in[0,1/2]$ in $\Sigma(\kappa)\times\R$ where $\kappa+4H^2\leq0$. The surfaces are invariant under $2\pi/k$-rotations about a vertical fiber of…

We show that constant mean curvature hypersurfaces in $\mathbb H^n\times\mathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces…

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to…

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 give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact…

This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension $3\le n+1 \le 7$ and consider a smooth function…

In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs…

For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution…

Let $M^4\to \mathbb{S}^5$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ in a $5$-dimensional sphere $\mathbb{S}^5$. In this paper, we prove that if $3$-mean curvature $H_3$ and the…

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important…

Given $H\in [0,\infty),$ some sufficient conditions for existence of CMC $H$ graphs with boundary in two parallel planes of $\mathbb{H}^2\times\mathbb{R}$ are presented. Height estimates for outwards-oriented CMC surfaces…

This is a revised version (minor changes and a deeper insight in the positive curvature case). We prove some Caccioppoli's inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant…

We study isoparametric hypersurfaces, whose principal curvatures are all constant, in the pseudo-Riemannian space forms. In this paper, we investigate two topics. Firstly, according to representations of Clifford algebras, we give a…

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…

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally…

The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface,…

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable…

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…