Sharp local scalar-curvature rigidity of spherical caps

Let $B(\delta )\subset {\mathbb{S}}^n$ be a geodesic ball with the round metric $\bar{g}$ of sectional curvature one, where $n\ge 3$ and $0<\delta <\pi \mathrm{/}2$. 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 $R(g)\ge n(n-1)$ and $H(g)\ge (n-1)\cot \delta$ 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.