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Ricci-pinched 3-Manifolds with second-fundamental-form-pinched Boundary
Abstract
We prove a boundary analogue of the rigidity theorem for Ricci-pinched three-manifolds under a superquadratic volume-growth assumption. More precisely, we show that a complete connected three-manifold with convex boundary, pinched Ricci tensor, and pinched second fundamental form is isometric to the Euclidean half-space. The proof is potential-theoretic. We construct and study mixed Dirichlet--Neumann p-capacitary potentials, establish monotonicity formulas for their free-boundary level sets, and use a weak Gauss--Bonnet formula adapted to this setting.
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BibTeX
@misc{xia2026riccipinched,
title = {{Ricci-pinched 3-Manifolds with second-fundamental-form-pinched Boundary}},
author = {Chao Xia and Jincong Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23261},
eprint = {2609.23261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23261},
}
RIS
TY - UNPB
TI - Ricci-pinched 3-Manifolds with second-fundamental-form-pinched Boundary
AU - Chao Xia
AU - Jincong Zheng
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23261
UR - https://arxiv.org/abs/2609.23261
AB - We prove a boundary analogue of the rigidity theorem for Ricci-pinched three-manifolds under a superquadratic volume-growth assumption. More precisely, we show that a complete connected three-manifold with convex boundary, pinched Ricci tensor, and pinched second fundamental form is isometric to the Euclidean half-space. The proof is potential-theoretic. We construct and study mixed Dirichlet--Neumann p-capacitary potentials, establish monotonicity formulas for their free-boundary level sets, and use a weak Gauss--Bonnet formula adapted to this setting.
ER -