Stability of Convex Disks
Differential Geometry
2023-11-08 v3 Analysis of PDEs
Metric Geometry
Abstract
We prove that topological disks with positive curvature and strictly convex boundary of large length are close to round spherical caps of constant boundary curvature in the Gromov-Hausdorff sense. This proves stability for a theorem of F. Hang and X. Wang, and can be viewed as an affirmative answer to a convex stability version of the Min-Oo Conjecture in dimension two. As an intermediate step, we obtain a compactness result for a Liouville-type PDE problem.
Keywords
Cite
@article{arxiv.2301.13130,
title = {Stability of Convex Disks},
author = {Hunter Stufflebeam},
journal= {arXiv preprint arXiv:2301.13130},
year = {2023}
}
Comments
19 pages, comments welcome! Version 2: Simplified proof in section 3.5, added prop. 2.1, improved exposition. Version 3: Greatly improved section 3.5, improved prop. 3.1 to have explicit constants, added additional exposition