Related papers: Stable hypersurfaces with constant higher order me…
In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial…
In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean…
A marginally outer trapped hypersurface is a generalization of minimal hypersurfaces originated from general relativity. We show a curvature estimate for stable marginally outer trapped hypersurfaces up to the free boundary satisfying a…
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their…
We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of…
We show stability of spherical caps (SCs) lying on a flat surface, where the motion is governed by the volume-preserving Mean Curvature Flow (MCF). Moreover, we introduce a dynamic boundary condition that models a line tension effect on the…
In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which…
We consider the local theory of constant mean curvature surfaces that satisfy one or two integrable boundary conditions and determine the corresponding potentials for the generalized Weierstrass representation.
It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1},…
In this paper, we study $4$-dimensional complete hypersurfaces with $w$-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for $4$-dimensional complete hypersurfaces with $w$-constant mean curvature.…
We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the…
We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces…
In this paper, we give the geometric meaning of hypersurfaces with constant mean curvature in a Finsler manifold by using volume preserving variation. Then we give the correspondence between principal curvatures of submanifolds by a…
We classify the $O(p)\times O(q)$-invariant constant mean curvature hypersurfaces with singularity at the origin, solving a conjecture of Wu-yi Hsiang.
We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When…
We apply the evolution method to present a new proof of the Alexandrov type theorem for constant anisotropic mean curvature hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$.
We generalize some results of Coray on closed points on cubic hypersurfaces. We show certain symmetric products of cubic hypersurfaces are stably birational.
We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…
In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show…
We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient…