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

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