Complete Constant Mean Curvature Research Articles

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.

On Calabi–Bernstein’s Problem for Uniqueness of Complete Spacelike Hypersurfaces in Weighted Generalized Robertson–Walker Spacetimes

In this paper, we study Calabi–Bernstein’s problem for the uniqueness of complete constant weighted mean curvature spacelike hypersurfaces Σn in weighted generalized Robertson–Walker spacetimes −I×ρMfn. Under appropriate geometric assumptions, we confirm that Σn is a spacelike slice.

New Bernstein Type Results in Weighted Warped Products

In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.

Complete constant mean curvature hypersurfaces in Euclidean space of dimension four or higher

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. One of the initial hypersurfaces is then perturbed to produce the desired CMC $n$-hypersurface which depends on the given family of perturbations of the graph and a small in absolute value parameter $\underline\tau$. This construction is very general because of the abundance of graphs which satisfy the required conditions and because it does not rely on symmetry requirements. For any given $k\ge2$ and $n\ge3$ it allows us to realize infinitely many topological types as CMC $n$-hypersurfaces in $\mathbb R^{n+1}$ with $k$ ends. Moreover for each case there is a plethora of examples reflecting the abundance of the available graphs. This is in sharp contrast with the known examples which in the best of our knowledge are all (generalized) cylindrical obtained by ODE methods and are compact or with two ends. Furthermore we construct embedded examples when $k\ge3$ where the number of possible topological types for each $k$ is finite but tends to $\infty$ as $k\to\infty$.

Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity

Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type $M^n\times_{\rho}\mathbb{R}_1$ whose Riemannian base $M^n$ has nonnegative sectional curvature and the warping function $\rho$ is convex on $M^n$, we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in $M^2\times_{\rho}\mathbb{R}_1$ and we obtain uniqueness results for entire Killing graphs constructed over $M^n$.

Calabi–Bernstein type results for complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes

In this article we study complete constant mean curvature spacelike hypersurfaces in spatially open Generalized Robertson–Walker spacetimes. Under appropriate geometric assumptions we prove new parametric uniqueness results as well as obtain very general Calabi–Bernstein type results for the constant mean curvature equation for spacelike graphs in these ambient spacetimes.

Chord arc properties for constant mean curvature disks

We prove a chord arc bound for disks embedded in $\mathbb{R}^3$ with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in $\mathbb{R}^3$ with finite topology or with positive injectivity radius.

Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality

We deal with complete constant mean curvature spacelike hypersurfaces immersed in a Lorentzian space form and satisfying a suitable Okumura type inequality, which is a weaker hypothesis than to assume that the spacelike hypersurface has two distinct principal curvatures. In this setting, we obtain estimates for the norm of the traceless part of the second fundamental form and we show that these estimates are sharp in the sense that the hyperbolic cylinders realize them.

On the first stability eigenvalue of hypersurfaces in the Euclidean and hyperbolic spaces

In this paper, we obtain upper bounds for the first eigenvalue of the stability operator of a closed constant mean curvature hypersurface ∑n immersed either in the Euclidean space ℝn+1 or in the hyperbolic space ℍn+1, n ≥ 2, in terms of the mean curvature and the length of the total umbilicity operator of ∑n. As application, we derive a nonexistence result concerning strong stable hypersurfaces in these ambient spaces. Furthermore, through the calculus of the first stability eigenvalue of circular cylinders in ℝn+1 and of hyperbolic cylinders in ℍn+1, we present a conjecture related to the first stability eigenvalue of complete constant mean curvature hypersurfaces immersed either in ℝn+1 or in ℍn+1.

Strongly stable surfaces in sub-Riemannian 3-space forms

A surface of constant mean curvature (CMC) equal to H in a sub-Riemannian 3-manifold is strongly stable if it minimizes the functional area+2Hvolume up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian 3-manifolds. We also produce new examples of C1 complete CMC surfaces with empty singular set in the sub-Riemannian 3-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic 3-space M(−1). In relation to the Bernstein problem in M(−1) we discover strongly stable C∞ entire minimal graphs in M(−1) different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class C2 and non-empty singular set in the sub-Riemannian 3-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).

Complete Constant Mean Curvature Spacelike Hypersurfaces in the Einstein-De Sitter Spacetime

The uniqueness and nonexistence results on complete constant mean curvature spacelike hypersurfaces lying between two spacelike slices in the Einstein–de Sitter spacetime are given. They are obtained from a Liouville-type theorem applied to a distinguished smooth function on a complete constant mean curvature hypersurface.

Embedded constant mean curvature surfaces in Euclidean three-space

In this paper we refine the construction and related estimates for complete constant mean curvature surfaces in Euclidean three-space developed in Kapouleas (Ann Math 131:239–330, 1990) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (Invent Math 119(3):443–518, 1995). As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in Kapouleas (Ann Math 131:239–330, 1990) and we produce a very large class of new embedded examples of finite topology.

Biharmonic Conformal Immersions into Three-Dimensional Manifolds

Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answer the question, “what surfaces can be biharmonically conformally immersed into Euclidean 3-space $${\mathbb{R}^3}$$ ?” We prove that a circular cylinder is the only CMC surface that can be biharmonically conformally immersed into $${\mathbb{R}^3}$$ ; we obtain a classification of biharmonic conformal immersions of complete CMC surfaces into $${\mathbb{R}^3}$$ and hyperbolic 3-spaces. We also study rotational surfaces that can be biharmonically conformally immersed into $${\mathbb{R}^3}$$ , and prove that a circular cone can never be biharmonically conformally immersed into $${\mathbb{R}^3}$$ .

On the umbilicity of complete constant mean curvature spacelike hypersurfaces

In this paper, we show that a complete spacelike hypersurface immersed with constant mean curvature either in the de Sitter space or in the anti-de Sitter space must be totally umbilical, provided that its Gauss mapping has some suitable behavior. In particular, we use an extension of Hopf’s maximum principle due to Yau in order to give new positive answers to Goddard’s conjecture.

On the angle of complete CMC hypersurfaces in Riemannian product spaces

We deal with complete two-sided hypersurfaces immersed with constant mean curvature in a Riemannian product space R×Mn. First, when the fiber Mn is compact with positive sectional curvature, we apply Bochner's technique jointly with Omori–Yau's generalized maximum principle to establish sufficient conditions which assure that such a hypersurface Σn is a slice {t}×Mn, provided that its (not necessarily constant) angle has some suitable boundedness. Afterwards, when Mn is complete with nonnegative sectional curvature, we proceed with our technique in order to guarantee that a complete constant mean curvature hypersurface immersed in R×Mn is minimal. Moreover, we characterize the uniqueness of entire vertical graphs Σn(u) with constant mean curvature into R×Mn, in terms of the norm of the gradient of the function u which determines such a graph. A nontrivial example of entire vertical graph with constant angle in R×H2 is also given, showing that our results do not hold when the fiber of the ambient product space has negative sectional curvature.

Constant Mean Curvature Spacelike Surfaces in Three-Dimensional Generalized Robertson–Walker Spacetimes

Several uniqueness and non-existence results on complete constant mean curvature spacelike surfaces lying between two slices in certain three-dimensional generalized Robertson–Walker spacetimes are given. They are obtained from a local integral estimation of the squared length of the gradient of a distinguished smooth function on a constant mean curvature spacelike surface, under a suitable curvature condition on the ambient spacetime. As a consequence, all the entire bounded solutions to certain family of constant mean curvature spacelike surface differential equations are found.

Ramification estimates for the hyperbolic Gauss map

We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.

Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 x R$

Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 x R$

Stable complete noncompact hypersurfaces with constant mean curvature

This article concerns the structure of complete noncompact stable hypersurfaces M n with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N n+1. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M n , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with \(H^{2} > \frac{65}{63},\frac{175}{148},\frac{41}{25},\frac{671}{171}\,{\rm for}\, n=3,4,5,6\), respectively, has only one end.