Sharp local scalar-curvature rigidity of spherical caps
Abstract
Let B(δ)⊂𝕊n be a geodesic ball with the round metric g̅ of sectional curvature one, where n≥3 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 C2,α perturbation g of g̅ satisfying R(g)≥ n(n−1) and H(g)≥(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 Rg̅(h)=D Hg̅(h)=0. The hemisphere counterexamples of Brendle--Marques--Neves show that the radius range is sharp.
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BibTeX
@misc{gong2026sharp,
title = {{Sharp local scalar-curvature rigidity of spherical caps}},
author = {Liuwei Gong and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29579},
eprint = {2610.29579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29579},
}
RIS
TY - UNPB
TI - Sharp local scalar-curvature rigidity of spherical caps
AU - Liuwei Gong
AU - Juncheng Wei
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29579
UR - https://arxiv.org/abs/2610.29579
AB - Let B(δ)⊂𝕊^n be a geodesic ball with the round metric g̅ of sectional curvature one, where n≥3 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 g̅ satisfying R(g)≥ n(n−1) and H(g)≥(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_g̅(h)=D H_g̅(h)=0. The hemisphere counterexamples of Brendle--Marques--Neves show that the radius range is sharp.
ER -