Height estimates and half-space theorems for hypersurfaces in product spaces of the type ℝ × M n.

Eudes L De Lima,Henrique F De Lima

doi:10.1590/0001-3765202120190329

Abstract

We obtain height estimates and half-space theorems concerning a wide class of hypersurfaces immersed into a product space ℝ × M n , the so-called generalized linear Weingarten hypersurfaces, which extends that one having some constant higher order mean curvature.

Highlights

The last decades have seen a steadily growing interest in the study of a priori estimates for the height function of constant mean curvature compact graphs or, more generally, compact hypersurfaces with boundary having some constant higher order mean curvature
More than twenty years after that, Korevaar et al 1992 obtained a sharp bound for compact graphs and for compact embedded hypersurfaces in the hyperbolic space Hn+1 with nonzero constant mean curvature and boundary in a horosphere
Given an arbitrary Riemannian manifold Mn, height estimates in the product space R × Mn for constant mean curvature compact embedded hypersurfaces with boundary in a slice were exhibited by Hoffman et al 2006 and Aledo et al 2008, for n = 2, and by Alías & Dajczer 2007, for an arbitrary dimension n

Summary

- INTRODUCTION

The last decades have seen a steadily growing interest in the study of a priori estimates for the height function of constant mean curvature compact graphs or, more generally, compact hypersurfaces with boundary having some constant higher order mean curvature. Given an arbitrary Riemannian manifold Mn, height estimates in the product space R × Mn for constant mean curvature compact embedded hypersurfaces with boundary in a slice were exhibited by Hoffman et al 2006 and Aledo et al 2008, for n = 2, and by Alías & Dajczer 2007, for an arbitrary dimension n. Cheng & Rosenberg 2005 were able to generalize these estimates for compact graphs with some constant higher order mean curvature in the product manifold R × Mn, with boundary in a slice.

Lr Θ

Lemma we have c

Proposition we get that s s