Rigidity for spin fill-ins with scalar curvature bounded from below

Abstract

We establish the rigidity statement in the equality case of the hyperspherical-radius inequality of Brendle, Tsiamis, and Wang for compact spin fill-ins with scalar curvature bounded below. More precisely, let $(M^{n\ge 3},g)$ be a compact, connected Riemannian spin manifold having a connected boundary $\Sigma$ and scalar curvature satisfying ${\mathrm{scal}}_g\ge -n(n-1)$. We prove that equality in the upper bound \[ \inf_ΣH\leq (n-1)\sqrt{1+\operatorname{Rad}(Σ)^{-2}} \] given by Brendle, Tsiamis, and Wang holds if and only if $(M,g)$ is isometric to a geodesic ball in hyperbolic space.