Constant Mean Curvature Research Articles

THE FORM OF THE COMPLETE SURFACES OF CONSTANT MEAN CURVATURE

We explained and classified the complete surfaces of constant mean curvature in addition to construct the first examples of complete surface of positive curvature, properly embedded minimal surfaces and we prove that every complete connected immersed surface with positive extrinsic curvature in must be properly embedded, homeomorphic to a sphere or a plane. We followed the analytical mathematical method and we found that the complete surface of positive curvature has multi applications in different fields of science specially in physics.

Surfaces of Osculating Circles in Euclidean Space

The aim of this paper is to define a new class of surfaces in Euclidean space using the concept of osculating circle. Given a regular curve C, the surface of osculating circles generated by C is the set of all osculating circles at all points of C. It is proved that these surfaces contain a one-parametric family of planar lines of curvature. A classification of surfaces of osculating circles is given in the family of canal surfaces, Weingarten surfaces, surfaces with constant Gauss curvature and surfaces with constant mean curvature.

On η-biharmonic hypersurfaces with constant scalar curvature in higher dimensional pseudo-Riemannian space forms

In this paper, η-biharmonic hypersurfaces Mrn with constant scalar curvature in a pseudo-Riemannian space form are studied. Under the assumption that Mrn has diagonalizable shape operator with at most six distinct principal curvatures, we prove that Mrn has constant mean curvature. As an application, we obtain a partial classification result of these hypersurfaces and show that (nc)-biharmonic hypersurfaces must be minimal.

Slab theorem and halfspace theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

We prove that a properly embedded annular end of a surface in \mathbb{H}^2\times\mathbb{R} with constant mean curvature 0<H\leq 1/2 can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature 0<H\leq 1/2 contained in \mathbb{H}^2 \times [0,+\infty) and with finite topology is necessarily a graph over a simply connected domain of \mathbb{H}^2 . For the case H=1/2 , the graph is entire.

First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere

In this paper, we study the first eigenvalue of the stability operator on an integral n n -varifold with constant mean curvature in the unit sphere S n + 1 \mathbb {S}^{n+1} . We find the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori.

On the Chern conjecture for isoparametric hypersurfaces

For a closed hypersurface Mn ⊂ Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, we show that if \({\rm{tr}}\left({{{\cal A}^k}} \right)\) are constants for k = 3, …, n − 1 and the shape operator \({\cal A}\) then M is isoparametric. The result generalizes the theorem of de Almeida and Brito (1990) for n = 3 to any dimension n, strongly supporting the Chern conjecture.

A Note on Constant Mean Curvature Foliations of Noncompact Riemannian Manifolds

We aimed to study constant mean curvature foliations of noncompact Riemannian manifolds, satisfying some geometric constraints. As a byproduct, we answer a question by M. P. do Carmo (see Introduction) about the leaves of such foliations.

A new gap for complete hypersurfaces with constant mean curvature in space forms

Let $M$ be an $n$-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in an unit sphere. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form respectively. We prove that if $S > \alpha (n, H)$, where $n\geq 4$ and $H\neq 0$, then $S > \alpha (n, H) + B_n\frac{n H^2}{n - 1}$. Here \[ \alpha (n, H) = n + \frac{n^3}{2 (n - 1)} H^2 - \frac{n (n - 2)}{2 (n - 1)}\sqrt{n^2 H^4 + 4 (n - 1) H^2}, \] $B_n=\frac{1}{5}$ for $4\leq n \leq 20$, and $B_n=\frac{49}{250}$ for $n>20$. Moreover, we obtain a gap theorem for complete hypersurfaces with constant mean curvature and constant scalar curvature in space forms.

Double bubbles with high constant mean curvatures in Riemannian manifolds

We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik–Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.

Special Slant Surfaces with Non-constant Mean Curvature in 2-Dimensional Complex Space Forms

In the late 1990s, B. Y. Chen introduced the notion of special slant surfaces in K\"{a}hler surfaces and classified non-minimal proper special slant surfaces with constant mean curvature in $2$-dimensional complex space forms. In this paper, we completely classify proper special slant surfaces with non-constant mean curvature in $2$-dimensional complex space forms.

Surfaces of prescribed linear Weingarten curvature in

Given $a,\,b\in \mathbb {R}$ and $\Phi \in C^{1}(\mathbb {S}^{2})$, we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb {R}^{3}$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi (N)$, where $N:\Sigma \rightarrow \mathbb {S}^{2}$ is the Gauss map. This theory widely generalizes some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.

Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

We show that a certain simply-stated notion of “analytic completeness” of the image of a real analytic map implies the map admits no analytic extension. We also give a useful criterion for that notion of analytic completeness by defining arc-properness of continuous maps, which can be considered as a very weak version of properness. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces (the so-called “G-catenoids”) or their analytic extensions in the de Sitter 3-space.

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.

Embedded Delaunay tori and their Willmore energy

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below 8π. Combining such a strict inequality with previous works by Keller–Mondino–Rivière and Mondino–Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.

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.

Rigidity of hypersurfaces with constant higher order mean curvature in space forms

Rigidity of hypersurfaces with constant higher order mean curvature in space forms

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.

Delaunay Surfaces of Prescribed Mean Curvature in $$\mathrm {Nil}_3$$ and $$\widetilde{SL_2}(\mathbb {R})$$

We obtain a classification result for rotational surfaces in the Heisenberg space and the universal cover of the special linear group, whose mean curvature is given as a prescribed \(C^1\) function depending on their angle function. We show that these surfaces behave like the Delaunay surfaces of constant mean curvature, under some assumptions on the prescribed function. In contrast with the constant mean curvature case, we exhibit the existence of rotational, embedded tori, providing counterexamples of the Alexandrov problem for this class of immersed surfaces.

A quasiconformal Hopf soap bubble theorem

We show that any compact surface of genus zero in {mathbb {R}}^3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon’s quasiconformal Bernstein theorem. The result generalizes, among others, Hopf’s theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.