Geometric barriers for the existence of hypersurfaces with prescribed curvatures in $M^n\times R$

TL;DR

This paper develops a deformation method for hypersurfaces in product spaces, enabling control over principal curvatures via the base manifolds' curvature, to establish existence or non-existence results for hypersurfaces with prescribed curvatures.

Contribution

It introduces a deformation process that relates hypersurfaces in different product spaces, providing a new tool for analyzing prescribed curvature problems.

Findings

Constructed barriers for hypersurfaces in product spaces.

Established conditions for existence and non-existence of prescribed curvature hypersurfaces.

Linked principal curvature relations to sectional and Ricci curvatures of base manifolds.

Abstract

We show the existence of a deformation process of hypersurfaces from a product space $M_1\times R$ into another product space $M_2\times R$ such that the relation of the principal curvatures of the deformed hypersurfaces can be controlled in terms of the sectional curvatures or Ricci curvatures of $M_1$ and $M_2$. In this way, we obtain barriers which are used for proving existence or non existence of hypersurfaces with prescribed curvatures in a general product space $M\times R$.