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

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