Curvature Hypersurfaces Research Articles

The Equality Case in the Substatic Heintze–Karcher Inequality

We provide a rigidity statement for the equality case of the Heintze–Karcher inequality in substatic manifolds. We apply such a result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle’s characterization of constant mean curvature hypersurfaces in warped products.

Local rigidity of constant mean curvature hypersurfaces in space forms

In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let x:Mn→Mn+1(c),n≥4, be a piece of immersed constant mean curvature hypersurface in the (n+1)-dimensional space form Mn+1(c). We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies g≤3, then Mn is an isoparametric hypersurface. Further, if Mn is a minimal hypersurface, then Mn is a totally geodesic hypersurface for c≤0, and Mn is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for c>0, which solves the high dimensional version of Bryant Conjecture.

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.

Thermodynamic Topology of Topological Black Hole in F(R)-ModMax Gravity’s Rainbow

Abstract In order to include the effect of high energy and topological parameters on black holes in $\mathrm{ F}(R)$ gravity, we consider two corrections to this gravity: energy-dependent spacetime with different topological constants, and a nonlinear electrodynamics field. In other words, we combine $\mathrm{ F}(R)$ gravity’s rainbow with ModMax nonlinear electrodynamics theory to see the effects of high energy and topological parameters on the physics of black holes. For this purpose, we first extract topological black hole solutions in $\mathrm{ F}(R)$-ModMax gravity’s rainbow. Then, by considering black holes as thermodynamic systems, we obtain thermodynamic quantities and check the first law of thermodynamics. The effect of the topological parameter on the Hawking temperature and the total mass of black holes is obvious. We also discuss the thermodynamic topology of topological black holes in $\mathrm{ F}(R)$-ModMax gravity’s rainbow using the off-shell free energy method. In this formalism, black holes are assumed to be equivalent to defects in their thermodynamic spaces. For our analysis, we consider two different types of thermodynamic ensembles. These are: fixed q ensemble and fixed $\phi$ ensemble. We take into account all the different types of curvature hypersurfaces that can be constructed in these black holes. The local and global topology of these black holes are studied by computing the topological charges at the defects in their thermodynamic spaces. Finally, in accordance with their topological charges, we classify the black holes into three topological classes with total winding numbers corresponding to $-1, 0$, and 1. We observe that the topological classes of these black holes are dependent on the value of the rainbow function, the sign of the scalar curvature, and the choice of ensembles.

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.

Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model

Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model

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

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.

Elliptic Weingarten hypersurfaces of Riemannian products

AbstractLet be either a simply connected space form or a rank‐one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of , which are those whose principal curvatures and angle function satisfy a relation , being W a differentiable function which is symmetric with respect to . When on the positive cone of , a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann‐type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either or is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of and which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of and .

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.

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.

Behavior of Prescribed Mean Curvature Hypersurfaces on Reentrant Ridges

Behavior of Prescribed Mean Curvature Hypersurfaces on Reentrant Ridges

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 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.

Minimal hypersurfaces with low index in the real projective space

In this paper we use suitable ideas from the min–max theory in finite dimension to prove that the only compact two-sided minimal and constant scalar curvature hypersurfaces in the real projective space with Morse index less or equal to two are the totally geodesic sphere or the Clifford minimal hypersurfaces.