On the volume of locally conformally flat 4 dimensional hypersphere

Abstract

Let $M$ be a 5 dimensional Riemannian manifold with $Sec_M\in[0,1]$, $\Sigma$ be a locally conformally flat hypersphere in $M$ with mean curvature $H$. We prove that, there exists $\varepsilon_0>0$, such that $\int_\Sigma (1+H^2)^2 \ge 8\pi^2/3$, provided $H \le \varepsilon_0$. In particular, if $\Sigma$ is a locally conformally flat minimal hypersphere in $M$, then $Vol(\Sigma) \ge 8\pi^2/3$, which partially answer a question proposed by Mazet and Rosenberg \cite{Ma&Rosen}. For an $(n+1)-$ dimensional rotationally symmetric Riemannian manifold $M$, we show that an immersed hypersurface $\Sigma$ is locally conformally flat if and only if ($n-1$) of the principal curvatures of $\Sigma$ are the same, which is a generalization of Cartan's result \cite{Cartan}. As an application, we prove that if $M$ is (some special but large class) rotationally symmetric 5-manifold with $Sec_M\in [0,1]$, and $\Sigma$ is a locally conformally flat hypersphere with mean curvature $H$, the inequality $\int_\Sigma (1+H^2)^2 \ge 8\pi^2/3$ holds for all $H$.