A M\"obius scalar curvature rigidity on compact conformally flat hypersurfaces in $\mathbb{S}^{n+1}$

TL;DR

This paper classifies conformally flat hypersurfaces in spheres with constant M"obius scalar curvature, showing that compact examples are M"obius equivalent to specific tori, revealing a rigidity phenomenon in M"obius geometry.

Contribution

It provides a complete classification of conformally flat hypersurfaces with constant M"obius scalar curvature in spheres, including explicit descriptions and a rigidity result for compact cases.

Findings

Compact conformally flat hypersurfaces with constant M"obius scalar curvature are M"obius equivalent to certain tori.

The scalar curvature for such hypersurfaces is explicitly given by R=(n-1)(n-2)r^2 with 0<r<1.

The classification explicitly characterizes all such hypersurfaces in terms of M"obius transformations.

Abstract

In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of M\"obius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant M\"obius scalar curvature under the M\"obius transformation group of $\mathbb{S}^{n+1}$. Second, we prove that if the conformally flat hypersurface with constant M\"obius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is M\"obius equivalent to the torus $$\mathbb{ S}^1(\sqrt{1-r^2})\times \mathbb{S}^{n-1}(r)\hookrightarrow \mathbb{S}^{n+1}.$$