arXiv research

No proper biharmonic CMC compact hypersurface in a specific warped product space.

The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…

The paper constructs all cmc hypersurfaces with two principal curvatures.

The paper proves that for a given metric, there exists another metric where the number of constant mean curvature hypersurfaces increases.

Unique CMC foliation in Minkowski space solved.

Improved gap for mean curvature of biharmonic hypersurfaces in spheres.

The paper proves properties of triharmonic CMC hypersurfaces with specific curvature conditions.

Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.

The paper studies triharmonic hypersurfaces in space forms and proves their properties.

The paper extends radius estimates for stable hypersurfaces in 2, 3, and 4 dimensions.

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving …

We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.

The paper proves properties of triharmonic CMC hypersurfaces with limited curvature types.

Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.

Authors review CMC spacelike hypersurfaces and new existence results.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

Proves properties of CMC hypersurfaces in specific spaces.

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}, different from an Euclidean sphere, must have stability index greater than or equa…

The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.

We define a Gauss map $γ:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $γ$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $γ$, are then obtained. By a…

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in $\R^{n}$ (Delaunay unduloids). When $n=3$, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfac…

Paper proves CMC hypersurfaces in R6 are minimal if they have finite index.

Constructs cmc doublings of minimal surfaces via min-max theory.

The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.

Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC hypersurfaces with optimal regularity. More quantitatively, there exists a consta…

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by glu…

Paper proves no specific CMC hypersurfaces in hyperbolic space.

Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.

The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.

We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We…

Paper classifies critical points in half-space with new distance function.

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tor…

It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature $c$. Moreover…

Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.

Study CMC hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with a specific symmetry.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

In this paper we study sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a…

Estimates bandwidth for CMC initial data sets.

New method improves curvature estimates for stable surfaces.

In this paper we consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric and derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the (2m+1)-dimensional Heisenberg gr…

The paper finds new constant mean curvature hypersurfaces in spheres.

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spa…

We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha in t…

The paper proves the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful charac…