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arXiv · 2610.29579

Sharp local scalar-curvature rigidity of spherical caps

Liuwei Gong·Juncheng Wei
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Abstract

Let $B(δ)\subset\mathbb S^n$ be a geodesic ball with the round metric $\bar{g}$ of sectional curvature one, where $n\geq3$ and $0<δ<π/2$. 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 $C^{2,α}$ perturbation $g$ of $\bar{g}$ satisfying $R(g)\geq n(n-1)$ and $H(g)\geq(n-1)\cotδ$ 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.

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Liuwei Gong, Juncheng Wei. 2026-10-21. Sharp local scalar-curvature rigidity of spherical caps. https://arxiv.org/abs/2610.29579

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