A remark on compact hypersurfaces with constant mean curvature in space forms

TL;DR

This paper characterizes compact hypersurfaces with constant mean curvature in space forms, showing they are either totally umbilical or locally rotational under certain conditions, extending classical results in differential geometry.

Contribution

It provides a new integral pinching condition that classifies such hypersurfaces, generalizing known theorems to higher dimensions and different curvature settings.

Findings

Hypersurfaces are either totally umbilical or locally rotational.

In dimension two, the condition reduces to a topological assumption.

The results extend classical theorems like Hopf-Chern.

Abstract

In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf-Chern result.