Scalar curvature rigidity of parabolically convex domains in hyperbolic spaces

For a parabolically convex domain $M\subseteq {\mathbb{H}}^n$, $n\ge 3$, we prove that if $f:(N,\bar{g})\to (M,g)$ has nonzero degree, where $N$ is spin with scalar curvature $R_N\ge -n(n-1)$, and if $f{\mathrm{\mid }}_{\mathrm{\partial }N}$ does not increase the distance and the mean curvature, then $N$ is hyperbolic, and $\mathrm{\partial }N$ is isometric to $\mathrm{\partial }M$. This is a partial generalization of Lott’s result \cite{lott2021index} to negative lower bounds of scalar curvature.