Jorge Herbert S. de Lira

Constant mean curvature surfaces in

The subject of this paper is properly embedded H-surfaces in Riemannian three manifolds of the form M 2 x R, where M 2 is a complete Riemannian surface. When M 2 = R 2 , we are in the classical domain of H-surfaces in R 3 . In general, we will make some assumptions about M 2 in order to prove stronger results, or to show the effects of curvature bounds in M 2 on the behavior of H-surfaces in M 2 x R.

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.

Geodesic Graphs with Constant Mean Curvature in Spheres

AbstractWe study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere

Height Estimates for Killing Graphs

\mathbb{S}^{n + 1} (a)

THE CHRISTOFFEL PROBLEM IN LORENTZIAN GEOMETRY

of radius a. Given a compact domain Ω, with some conditions, contained in a totally geodesic sphere S of

Helicoidal graphs with prescribed mean curvature

\mathbb{S}^{n + 1} (a)