Scalar curvature rigidity of parabolically convex domains in hyperbolic spaces

TL;DR

This paper proves a rigidity result for scalar curvature in hyperbolic spaces, showing that certain maps with nonzero degree imply the domain must be hyperbolic with boundary isometric to the original domain.

Contribution

It generalizes Lott's scalar curvature rigidity result to cases with negative scalar curvature bounds in hyperbolic spaces.

Findings

N is hyperbolic under given conditions

Boundary of N is isometric to boundary of M

Results extend scalar curvature rigidity to negative bounds

Abstract

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 $f|_{\partial N}$ does not increase the distance and the mean curvature, then $N$ is hyperbolic, and $\partial N$ is isometric to $\partial M$. This is a partial generalization of Lott's result \cite{lott2021index} to negative lower bounds of scalar curvature.