On complete hypersurfaces with constant scalar curvature $n(n-1)$ in the unit sphere

TL;DR

This paper proves that complete, locally conformally flat hypersurfaces in the sphere with constant scalar curvature are totally geodesic if their total mean curvature is sufficiently small.

Contribution

It establishes a new rigidity result for hypersurfaces with constant scalar curvature under a small total curvature condition.

Findings

Hypersurfaces with small total mean curvature are totally geodesic.

The result applies to complete, locally conformally flat hypersurfaces in the sphere.

The condition on total curvature is sufficient for rigidity.

Abstract

Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v \right) ^ { \frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.