On the volume of locally conformally flat 4 dimensional hypersphere

TL;DR

This paper establishes volume bounds for locally conformally flat hyperspheres in certain 5-dimensional manifolds, extending classical results and addressing a question by Mazet and Rosenberg, with implications for minimal hyperspheres.

Contribution

It proves a volume inequality for locally conformally flat hyperspheres with small mean curvature and characterizes conformally flat hypersurfaces in rotationally symmetric manifolds, generalizing Cartan's theorem.

Findings

Volume bound for hyperspheres with small mean curvature in 5-manifolds.

Characterization of conformally flat hypersurfaces in rotationally symmetric manifolds.

Extension of classical results to higher dimensions and specific curvature conditions.

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 $H$.