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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 arotationally symmetric and weakly convex body $\bar{M}$ in $\mathbb{R}^3$. Wecall $(\bar{M}, \delta)$ 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 $\partial M$ areno greater than the model, then $M$ is flat. We also generalize this result tothe hyperbolic case and some spaces with $\mathbb{S}^1$-symmetry. Our approachis inspired by Gromov.

Scalar curvature comparison of rotationally symmetric sets