Sharp local scalar-curvature rigidity of spherical caps

Abstract

Let $B(\delta)\subset\mathbb S^n$ be a geodesic ball with the round metric $\bar{g}$ of sectional curvature one, where $n\geq3$ and $0<\delta<\pi/2$. We prove local scalar-curvature rigidity under preservation of the induced boundary metric and a one-sided lower bound for the boundary mean curvature. More precisely, every sufficiently small $C^{2,\alpha}$ perturbation $g$ of $\bar{g}$ satisfying $R(g)\geq n(n-1)$ and $H(g)\geq(n-1)\cot\delta$ is round up to a diffeomorphism fixing the boundary pointwise. Inspired by the scalar--vector--tensor decomposition used in the study of Yamabe compactness, we establish positivity of the Brendle--Marques quadratic form in the divergence-free gauge for variations preserving the induced boundary metric and satisfying the joint linearized constraints $D R_{\bar{g}}(h)=D H_{\bar{g}}(h)=0$. The hemisphere counterexamples of Brendle--Marques--Neves show that the radius range is sharp.