Scalar curvature rigidity of small spherical caps with arbitrary fill-ins

Abstract

We prove an analogue of Min-Oo's conjecture for sufficiently small spherical caps in the unit round sphere. For every dimension $n\ge 3$, there exists $r_{\ast }(n)>0$ such that the following holds for every round cap of radius $0<r\le r_{\ast }(n)$: any smooth compact connected oriented Riemannian fill-in with the same intrinsic boundary metric and with scalar curvature and outward boundary mean curvature at least those of the cap is isometric to the cap.