On the volume of locally conformally flat 4-dimensional closed hypersurface

Abstract

Let $M$ be a 5-dimensional Riemannian manifold with $Sec_M\in [0,1]$ and $\Sigma$ be a locally conformally flat closed hypersurface in $M$ with mean curvature function $H$. We prove that there exists $\varepsilon _0>0$ such that \begin{align} \int _\Sigma (1+H^2)^2 \ge \frac {4\pi ^2}{3}\chi (\Sigma ), \end{align} provided $\vert H\vert \le \varepsilon _0$, where $\chi (\Sigma )$ is the Euler number of $\Sigma$. In particular, if $\Sigma$ is a locally conformally flat minimal hypersphere in $M$, then $Vol(\Sigma ) \ge 8\pi ^2/3$, which partially answers a question proposed by Mazet and Rosenberg. Moreover, we show that if $M$ is (some special but large class) rotationally symmetric, then the inequality (\ref{V1}) holds for all $H$.