Luis J. Alías

Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes

A new technique is introduced in order to solve the following question:When is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice? (Generalized Robertson-Walker spacetimes extend classical Robertson-Walker ones to include the cases in which the fiber has not constant sectional curvature.) First, we determine when this hypersurface must be compact. Then, all these compact hypersurfaces in (necessarily spatially closed) spacetimes are shown to be totally umbilical and, except in very exceptional cases, slices. This leads to proof of a new Bernstein-type result. The power of the introduced tools is also shown by reproving and extending several known results.

INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their r-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their r-mean curvatures are linearly related or one of them is constant.

Björling problem for maximal surfaces in Lorentz–Minkowski space

We introduce a new approach to the local study of maximal surfaces in Lorentz–Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Bjorling-type problem in Lorentz–Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces.

CONSTANT HIGHER-ORDER MEAN CURVATURE HYPERSURFACES IN RIEMANNIAN SPACES

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R^3. In a recent paper the first and third authors have shown that this is true for the case of hypersurfaces in R^{n+1} with constant scalar curvature, and more generally, hypersurfaces with constant higher order r-mean curvature, when r>1. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold, where we will consider a general geometric configuration consisting of an immersed hypersurface with boundary on an oriented hypersurface P. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of P, as well as the geometry of P. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature. In particular, we are able to extend the previous symmetry results to the case of hypersurfaces with constant higher order r-mean curvature in the hyperbolic space and in the sphere.

CONSTANT MEAN CURVATURE HYPERSURFACES IN WARPED PRODUCT SPACES

We study hypersurfaces of constant mean curvature immersed into warped product spaces of the form

Integral formulas for compact space-like hypersurfaces in de Sitter space: Applications to the case of constant higher order mean curvature

\mathbb{R}\times_\varrho\mathbb{P}^n

Spacelike hypersurfaces with constant mean curvature in the steady state space

, where

Hypersurfaces of constant higher order mean curvature in warped products

\mathbb{P}^n

Maximum principles and geometric applications

is a complete Riemannian manifold. In particular, our study includes that of constant mean curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant mean curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation

On the Gaussian Curvature of Maximal Surfaces and the Calabi–Bernstein Theorem

t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n