Complete Hypersurfaces Research Articles

Higher order mean curvature estimates for bounded complete hypersurfaces

We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.

RIGIDITY THEOREMS IN THE HYPERBOLIC SPACE

As a suitable application of the well known generalized max- imum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the (n+ 1)-dimensional hyperbolic space Hn+1. In our approach, we explore the existence of a natural duality between Hn+1 and the half Hn+1 of the de Sitter space S n+1 1 , which models the so-called steady state space.

On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker spacetime and for hypersurfaces in hyperbolic space are given.

On Bernstein-Type Theorems in Semi-Riemannian Warped Products

Complete spacelike hypersurfaces immersed in semi-Riemannian warped products are investigated. By using a technique according to Yau (1976) and a reasonable restriction on the mean curvature of the hypersurfaces, we obtain some new Bernstein-type theorems which extend some known results proved by Camargo et al. (2011) and Colares and Lima (2012).

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\), provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.

On the Unicity of Complete Hypersurfaces Immersed in a Semi-Riemannian Warped Product

Our purpose is to apply appropriate generalized maximum principles in order to study the unicity of complete hypersurfaces immersed in a semi-Riemannian warped product, which is supposed to obey a suitable convergence condition. In this setting, by assuming a natural comparison inequality between the r-th mean curvatures of the hypersurface and the ones of the slices of the slab where the hypersurface is contained, we establish sufficient conditions to guarantee that such a hypersurface must be a slice.

Hypersurfaces of constant higher order mean curvature in warped products

In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.

Curvature flow of complete hypersurfaces in hyperbolic space

In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C 2 estimates.

On Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

We apply the well know Omori–Yau generalized maximum principle (Omori in J Math Soc Jpn 19:205–214, 1967; Yau in Commun Pure Appl Math 28:201–228, 1975), as well as a suitable extension of it that was established in a joint work with Caminha (Caminha and de Lima in Gen Relat Grav 41:173–189, 2009), in order to investigate Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey a standard convergence condition.

Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.

Bernstein theorems for complete α-relative extremal hypersurfaces

Let \({y : M \rightarrow \mathbb{R}^{n+1}}\) be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space \({\mathbb{R}^{n+1}}\), given as graph of a strictly convex function xn+1 = f(x1, . . . ,xn) defined on a domain \({\Omega \subset \mathbb{R}^{n}}\). Considering the Li-normalization of the graph of the convex function f, we will prove the Bernstein theorems for relative extremal hypersurfaces with complete α-metrics. As one case of main theorem, we obtain that affine complete maximal surface given by graph is an elliptic paraboloid, which is a special case of Calabi conjecture about affine maximal surfaces.

Some rigidity theorems in semi-Riemannian warped products

We study the problem of uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm. By applying a maximum principle at the infinity due to S. T. Yau and supposing a natural comparison inequality between the mean curvature of the hypersurface and that of the slices of the region where the hypersurface is contained, we obtain rigidity theorems in such ambient spaces. Applications to the hyperbolic and the steady state spaces are given.

LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACE

Let M be a linear Weingarten spacelike hypersurface in a locally symmetric Lorentz space with R = aH + b, where R and H are the normalized scalar curvature and the mean curvature, respectively. In this paper, we give some conditions for the complete hypersurface M to be totally umbilical.

A Bernstein type theorem in ℝ × ℍ n

In this paper, as suitable application of the so-called Omori-Yau generalized maximum principle, we obtain a Bernstein type theorem concerning to complete hypersurfaces immersed with constant mean curvature in the product space ℝ × ℍn. Furthermore, we treat the case that such hypersurfaces are vertical graphs.

Curvature Flow of Complete Convex Hypersurfaces in Hyperbolic Space

We investigate the existence, convergence, and uniqueness of modified general curvature flow (MGCF) of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space \({H^{n+1}_1(-1)}\). we prove that if a complete space-like hypersurface with constant mean curvature \({x:\mathbf M\rightarrow H^{n+1}_1(-1) }\) has two distinct principal curvatures λ, μ, and inf|λ − μ| > 0, then x is the standard embedding \({ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}\) in anti-de Sitter space \({ H^{n+1}_1 (-1) }\).

UNIQUENESS OF COMPLETE HYPERSURFACES WITH BOUNDED HIGHER ORDER MEAN CURVATURES IN SEMI-RIEMANNIAN WARPED PRODUCTS

AbstractIn this paper, we deal with complete hypersurfaces immersed with bounded higher order mean curvatures in steady state-type spacetimes and in hyperbolic-type spaces. By applying a generalised maximum principle for the Yau's square operator [11], we obtain uniqueness results in each of these ambient spaces.

Complete hypersurfaces immersed in a semi-Riemannian warped product

The aim of this paper is to study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice.

Erratum: On the Dirichlet problem for minimal graphs in hyperbolic space

R n × {0} that can be identified with the asymptotic infinity of Hn+1. Let Nn−1 ⊂ Hn+1 ∞ be a closed embedded (n − 1)-dimensional submanifold. Then there exists a complete absolutely area-minimizing integral n-current Σ asymptotic to Nn−1 at infinity; see, for example [1, 2]. It was also noted in [1, 2] that in the case n≤ 6, currents obtained are smoothly embedded complete hypersurfaces. In the case n ≥ 7, as in the Euclidean case, there can be a singular set of Hausdorff dimension at most n− 7.

Pinching theorems of hypersurfaces in a unit sphere

Let M n be a complete hypersurface in S n + 1 ( 1 ) with constant mean curvature. Assume that M n has n − 1 principal curvatures with the same sign everywhere. We prove that if Ric M ≤ C − ( H ) , either S ⩽ S + ( H ) or Ric M ⩾ 0 or the fundamental group of M n is infinite, then S is constant, S = S + ( H ) and M n is isometric to a Clifford torus S 1 ( 1 − r 2 ) × S n − 1 ( r ) with r 2 ⩾ n − 1 n . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature.