Constant Mean Curvature Hypersurfaces Research Articles

On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold

AbstractWe propose a notion of stability for constant ‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.

Umbilicity of constant mean curvature hypersurfaces into space forms

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.

Constant Mean Curvature Hypersurfaces in Anti-de Sitter Space

Abstract We study entire spacelike constant mean curvature hypersurfaces in Anti-de Sitter space of any dimension. First, we give a classification result with respect to their asymptotic boundary, namely we show that every admissible sphere $\Lambda $ is the boundary of a unique such hypersurface, for any given value $H$ of the mean curvature. We also demonstrate that, as $H$ varies in $\mathbb {R}$, these hypersurfaces analytically foliate the invisible domain of $\Lambda $. Finally, we extend Cheng-Yau Theorem to the Anti-de Sitter space, which establishes the completeness of any entire constant mean curvature hypersurface.

A survey about a Do Carmo’s question on complete stable constant mean curvature hypersurfaces

A survey about a Do Carmo’s question on complete stable constant mean curvature hypersurfaces

On the Normal Stability of Triharmonic Hypersurfaces in Space Forms

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere phi :mathbb {S}^m(1/sqrt{3})hookrightarrow mathbb {S}^{m+1} is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.

Stability of hypersurfaces with constant mean curvature trapped between two parallel hyperplanes

Static equilibrium configurations of continua supported by surface tension are given by constant mean curvature (CMC) surfaces which are critical points of a variational problem to extremize the area while keeping the volume fixed. CMC surfaces are used as mathematical models of a variety of continua, such as tiny liquid drops, stars, and nuclei, to play important roles in both mathematics and physics. Therefore, the geometry of CMC surfaces and their properties such as stability are of special importance in differential geometry and in a variety of physical sciences. In this paper we examine the stability of CMC hypersurfaces in arbitrary dimensions, possibly having boundaries on two parallel hyperplanes, by investigating the second variation of the area. We determine the stability of non-uniform liquid bridges or unduloids for the first time in all dimensions and all parameter (the ratio of the neck radius to bulge radius) regimes. The analysis is assisted by numerical computations.

Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi–Bernstein-type problems

In this article spacelike hypersurfaces immersed in twisted product spacetimes Itimes _f F with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, and a relation that needs to hold on each immersed spacelike hypersurface in Itimes _f F to be a simple warped product is also derived. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersurfaces and other for totally umbilic ones with a suitable restriction on their mean curvature are presented, and a full description of maximal hypersurfaces in twisted product spacetimes with a one-dimensional Lorentzian fiber is also included. Finally, the mean curvature equation for a spacelike graph on the fiber is computed and as an application, some Calabi–Bernstein-type results are proven.

Constant mean curvature hypersurfaces in $\mathbb{H}^n\times\mathbb{R}$ with small planar boundary

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 determined by $P$. This is the analogous in ${\mathbb H}^n\times\mathbb{R}$ of a result in $\mathbb{R}^3$ by A. Ros and H. Rosenberg \[J. Differential Geom. 44 (1996), 807–817].

Harmonic Gauss maps of submanifolds of arbitrary codimension of the Euclidean space and sphere and some applications

AbstractIt is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too.

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $$\mathbb {T}^N$$ starting near a strictly stable critical set E of the perimeter converges in the long time to a translate of E exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

Stability of hypersurfaces of constant mean curvature with free boundary in two parallel hyperplanes

Stability of hypersurfaces of constant mean curvature with free boundary in two parallel hyperplanes

The Intrinsic Structure of High-Dimensional Data According to the Uniqueness of Constant Mean Curvature Hypersurfaces

In this paper, we study the intrinsic structures of high-dimensional data sets for analyzing their geometrical properties, where the core message of the high-dimensional data is hiding on some nonlinear manifolds. Using the manifold learning technique with a particular focus on the mean curvature, we develop new methods to investigate the uniqueness of constant mean curvature spacelike hypersurfaces in the Lorentzian warped product manifolds. Furthermore, we extend the uniqueness of stochastically complete hypersurfaces using the weak maximum principle. For the more general cases, we propose some non-existence results and a priori estimates for the constant higher-order mean curvature spacelike hypersurface.

Hypersurfaces of constant higher-order mean curvature in $$M\times {\mathbb{R}}$$

We consider hypersurfaces of products $$M\times {\mathbb{R}}$$ with constant rth mean curvature $${H_{r}} \ge 0$$ (to be called $${{H}_{r}}$$ -hypersurfaces), where M is an arbitrary Riemannian n-manifold. We develop a general method for constructing them and employ it to produce many examples for a variety of manifolds M, including all simply connected space forms and the hyperbolic spaces $${\mathbb{H}}_{{\mathbb{F}}}^{m}$$ (rank one symmetric spaces of noncompact type). We construct and classify complete rotational $${{H}_{r}}(\ge 0)$$ -hypersurfaces in $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ and in $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ as well. They include spheres, Delaunay-type annuli and, in the case of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}},$$ entire graphs. We also construct and classify complete $${{H}_{r}}(\ge 0)$$ -hypersurfaces of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and strictly convex $${{H}_{r}}$$ -hypersurface of $${\mathbb{H}}^{n}\times {\mathbb{R}}$$ or $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ $$(n\ge 3)$$ is a rotational embedded sphere. Other uniqueness results for complete $${{H}_{r}}$$ -hypersurfaces of these ambient spaces are obtained.

The existence of G-invariant constant mean curvature hypersurfaces

In this paper, we consider a closed Riemannian manifold \(M^{n+1}\) with dimension \(3\le n+1\le 7\), and a compact Lie group G acting as isometries on M with cohomogeneity at least 3. Suppose the union of non-principal orbits \(M{\setminus } M^{reg}\) is a smooth embedded submanifold of M with dimension at most \(n-2\). Then for any \(c\in \mathbb {R}\), we show the existence of a nontrivial, smooth, closed, almost embedded, G-invariant hypersurface \(\Sigma ^n\) of constant mean curvature c.

A partially overdetermined problem in domains with partial umbilical boundary in space forms

Abstract In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings.

Constant weighted mean curvature hypersurfaces in shrinking Ricci solitons

In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a constant weighted mean curvature hypersurface in the Gaussian shrinking Ricci soliton (not necessarily properly immersed) with a certain assumption on the integral of the second fundamental form must be a generalized cylinder.

CMC hypersurfaces with bounded Morse index

Abstract We provide qualitative bounds on the area and topology of separating constant mean curvature (CMC) surfaces of bounded (Morse) index. We also develop a suitable bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular, we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids.

CMC Doublings of Minimal Surfaces via Min–Max

Let \(\Sigma ^2 \subset M^3\) be a minimal surface of index 0 or 1. Assume that a neighborhood of \(\Sigma \) can be foliated by constant mean curvature (cmc) hypersurfaces. We use min–max theory and the catenoid estimate to construct \(\varepsilon \)-cmc doublings of \(\Sigma \) for small \(\varepsilon > 0\). Such cmc doublings were previously constructed for minimal hypersurfaces \(\Sigma ^n \subset M^{n+1}\) with \(n+1\ge 4\) by Pacard and Sun using gluing methods.

Complete Non-compact Hypersurfaces of Constant Higher Order Mean Curvature in Lorentzian Warped Product Manifolds

Complete Non-compact Hypersurfaces of Constant Higher Order Mean Curvature in Lorentzian Warped Product Manifolds

The principal curvature theorem and its applications to constant mean curvature hypersurfaces in Euclidean space

The principal curvature theorem and its applications to constant mean curvature hypersurfaces in Euclidean space