Scalar curvature comparison of rotationally symmetric sets

Let $(M, g)$ be a compact 3-manifold with nonnegative scalar curvature $R_g\geq 0$. 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 $H_{\partial M} \geq \bar{H}_{\partial M}$, $\sigma \geq \bar{\sigma}$ 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.