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

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 $$\mathbb{ S}^1(\sqrt{1-r^2})\times \mathbb{S}^{n-1}(r)\hookrightarrow \mathbb{S}^{n+1}.$$