On the volume of locally conformally flat 4 dimensional hypersphere

Abstract

Let $M$ be a 5 dimensional Riemannian manifold with $Se{c}_M\in [0,1]$, $\hat{I}\pounds$ be a locally conformally flat hypersphere in $M$ with mean curvature $H$. We prove that, there exists ${\epsilon }_0>0$, such that ${\int }_{\hat{I}}\pounds (1+H^2{)}^2\ge 8\ddot{I}{€}^2/3$, provided $H\le {\epsilon }_0$. In particular, if $\hat{I}\pounds$ is a locally conformally flat minimal hypersphere in $M$, then $Vol(\hat{I}\pounds )\ge 8\ddot{I}{€}^2/3$, 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 $\hat{I}\pounds$ is locally conformally flat if and only if ($n-1$) of the principal curvatures of $\hat{I}\pounds$ 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 $Se{c}_M\in [0,1]$, and $\hat{I}\pounds$ is a locally conformally flat hypersphere with mean curvature $H$, the inequality ${\int }_{\hat{I}}\pounds (1+H^2{)}^2\ge 8\ddot{I}{€}^2/3$ holds for all $H$.

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