Related papers: A note on surfaces with parallel mean curvature

In this paper, we give a Simons' type formula for the cmc surfaces in homogeneous $3$-manifolds $E(\kappa,\tau)$, $\tau\neq0$. As an application, we give a rigidity result in the case of $\kappa> 4\tau^2$ for the cmc surfaces under a…

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces.…

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics…

This article surveys results for Riemannian manifolds of positive and non-negative sectional curvature with symmetries.

The aim of this paper is to extend classic results of the theory of CMC surfaces in the product spaces to the class of immersed surfaces in $\mathbb{M}^2(\kappa)\times\mathbb{R}$ whose mean curvature is given as a $C^1$ function depending…

A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…

In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the…

In this paper, we establish some modified defect relations for the Gauss map $g$ of a complete minimal surface $S\subset\mathbb R^m$ into $\mathbb P^n(\mathbb C)\ (n=m-1)$ with only a single Fermat hypersurface $Q$ of $\mathbb P^n(\mathbb…

We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of…

We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension $2n+1$). We prove a codimension reduction theorem. We…

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean…

Given a $C^1$ function $\mathcal{H}$ defined in the unit sphere $\mathbb{S}^2$, an $\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\mathcal{H}(N_p)$, $p\in M$, where…

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the…

We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a…

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean…

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature…

We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature…

This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a…