On the scalar curvature of constant mean curvature hypersurfaces in space forms

TL;DR

This paper investigates the scalar curvature of complete constant mean curvature hypersurfaces in space forms, providing a sharp estimate for its infimum using a maximum principle, advancing understanding of geometric properties in these settings.

Contribution

It introduces a sharp estimate for the infimum of scalar curvature on such hypersurfaces, applying a weak Omori-Yau maximum principle to derive new geometric bounds.

Findings

Derived a sharp estimate for the infimum of scalar curvature

Applied the weak Omori-Yau maximum principle in this context

Enhanced understanding of scalar curvature behavior in space form hypersurfaces

Abstract

In this paper we study the behavior of the scalar curvature $S$ of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of $S$. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli and Setti \cite{PRS}.