Related papers: Multi-valued graphs in embedded constant mean curv…

We prove that the Gauss map of a surface of constant mean curvature embedded in Minkowski space is harmonic. This fact will then be used to study 2+1 gravity for surfaces of genus higher than one. By considering the energy of the Gauss map,…

We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.

We prove that a closed immersed plane curve with total curvature $2\pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct…

We obtain necessary conditions for the existence of complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. In the two-dimensional case we prove Bernstein-type results in each of these ambient spaces.

We determine the local geometric structure of two-dimensional metric spaces with curvature bounded above as the union of finitely many properly embedded/branched immersed Lipschitz disks. As a result, we obtain a graph structure of the…

We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then…

We show that a properly immersed minimal hypersurface in M x R_+ equals some M x {c} when M is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. If on the other hand, M has nonnegative Ricci curvature with…

In this paper, using heat kernel estimates and contraction mapping principle, we give a new proof of the existence and uniqueness of mean curvature flow starting from hypersurface with bounded second fundamental form. Moreover, we show the…

An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full…

We consider a system of three surfaces, graphs over a bounded domain in ${\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to…

We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.

Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant…

We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.

In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere $\mathbb{S}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an…

We show that, for every n and every surface $\Sigma$, there is a graph U embeddable on $\Sigma$ with at most cn^2 vertices that contains as minor every graph embeddable on $\Sigma$ with n vertices. The constant c depends polynomially on the…

This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some…

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature.…

A sequence of constant mean curvature surfaces $\Sigma_j$ with mean curvature $H_j \to \infty$ in a three-dimensional manifold $M$ condenses to a compact and connected graph $\Gamma$ consisting of a finite union of curves if $\Sigma_j$ is…

In the Minkowski space, we consider a compact, spacelike hypersurface with boundary, which can be written as a graph on a spacelike hyperplane. We prove that, if its $k$-th mean curvature is constant, and its boundary is on the hyperplane…

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…