Stability of Convex Disks
arXiv:2301.13130 · doi:10.1007/s00526-023-02582-8
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.
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