Hypersurfaces with prescribed angle function

Henrique F. de Lima, Eraldo A. Lima Jr. and Ulisses L. Parente

Vol. 269 (2014), No. 2, 393–406

DOI: 10.2140/pjm.2014.269.393

Abstract

We deal with two-sided complete hypersurfaces immersed in a Riemannian product space, whose base is assumed to have sectional curvature bounded from below. In this setting, we obtain sufficient conditions which assure that such a hypersurface is a slice of the ambient space, provided that its angle function has some suitable behavior. Furthermore, we establish a natural relation between our results and the classical problem of describing the geometry of a hypersurface immersed in the Euclidean space through the behavior of its Gauss map.

Keywords

Riemannian product spaces, two-sided complete hypersurfaces, mean curvature, height function, entire vertical graphs

Mathematical Subject Classification 2010

Primary: 53C42

Secondary: 53C50, 53B30

Milestones

Received: 30 April 2013

Revised: 28 February 2014

Accepted: 4 March 2014

Published: 26 July 2014