Scalar curvature comparison of rotationally symmetric sets

Abstract

Let $(M,g)$ be a compact 3-manifold with nonnegative scalar curvature $R_g\ge 0$. The boundary $\partial M$ is diffeomorphic to the boundary of a rotationally symmetric and weakly convex body $\bar{M}$ in ${\mathbb{R}}^3$. We call $(\bar{M},\delta )$ a model or a reference. Let $H_{\partial M}$ and ${\bar{H}}_{\partial M}$ be respectively the mean curvatures of $\partial M$ in $(M,g)$ and $\partial M$ in $(\bar{M},\delta )$, $\sigma$ and $\bar{\sigma }$ be the induced metric from $g$ and $\delta$. We show that for some classes of $\partial M$, if $H_{\partial M}\ge {\bar{H}}_{\partial M}$, $\sigma \ge \bar{\sigma }$ and the dihedral angles at the nonsmooth part of $\partial M$ are no greater than the model, then $M$ is flat. We also generalize this result to the hyperbolic case and some spaces with ${\mathbb{S}}^1$-symmetry. Our approach is inspired by Gromov.

The assumption $g\ge \delta$ is not needed in the case (c) of Theorem 1.2 with unchanged proof. We have removed the assumption in the version submitted to journals, but failed to update the arXiv promptly