Scalar curvature rigidity of parabolically convex domains in hyperbolic spaces
Chengzhang Sun
TL;DR
The paper proves a scalar curvature rigidity result in hyperbolic space for parabolically convex domains: if a nonzero-degree map $f:(N,\bar{g})\to (M,g)$ from a spin manifold $N$ with $R_N\ge -n(n-1)$ preserves boundary mean curvature and is Lipschitz on the boundary, then $N$ must be hyperbolic with $\partial N$ isometric to $\partial M$, and the boundary map is an isometry on components. The authors develop a spinorial framework with a modified connection $\widehat{\nabla}$ and a twisted Dirac operator, derive a Lichnerowicz-type formula, and formulate a boundary-value problem whose solutions yield Killing spinors, constraining curvature to the hyperbolic model. The core argument combines boundary estimates for the modified Dirac operator with index theory to leverage the degree of $f$, handling odd and even dimensions via ungraded and graded constructions, respectively. This work generalizes scalar curvature rigidity to negative lower bounds and extends Lott-type results by connecting topological degree, boundary data, and interior hyperbolicity in the parabolic setting. The results advance our understanding of how global topological data and boundary convexity interact with curvature in hyperbolic geometry, offering a robust spinorial route to rigidity in negatively curved contexts.
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.