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Stability and uniqueness of minimal disks in non-constant curvature
Abstract
Nitsche proved that every smooth Jordan curve in ℝ3 of total curvature at most 4π bounds a unique minimal disk, which is moreover strictly stable. We prove an analogue of this result for Riemannian $3$-balls with mean convex boundary, under an explicit pinching condition on the negative sectional curvature, together with a bound on the covariant derivative of the Ricci tensor. In this setting, every smooth Jordan curve in the boundary sphere of total curvature at most 4π bounds a unique embedded minimal disk which is strictly stable.
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BibTeX
@misc{alvarez2026stability,
title = {{Stability and uniqueness of minimal disks in non-constant curvature}},
author = {Sébastien Alvarez and Thibault Lefeuvre and Ben Lowe and Graham A. Smith},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29542},
eprint = {2609.29542},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29542},
}
RIS
TY - UNPB
TI - Stability and uniqueness of minimal disks in non-constant curvature
AU - Sébastien Alvarez
AU - Thibault Lefeuvre
AU - Ben Lowe
AU - Graham A. Smith
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29542
UR - https://arxiv.org/abs/2609.29542
AB - Nitsche proved that every smooth Jordan curve in ℝ^3 of total curvature at most 4π bounds a unique minimal disk, which is moreover strictly stable. We prove an analogue of this result for Riemannian $3$-balls with mean convex boundary, under an explicit pinching condition on the negative sectional curvature, together with a bound on the covariant derivative of the Ricci tensor. In this setting, every smooth Jordan curve in the boundary sphere of total curvature at most 4π bounds a unique embedded minimal disk which is strictly stable.
ER -