Stability of Convex Disks

TL;DR

This paper demonstrates that convex disks with positive curvature and large boundary length are geometrically close to spherical caps, confirming a stability version of a known geometric theorem and addressing a convex stability aspect of the Min-Oo Conjecture in two dimensions.

Contribution

It establishes a stability result for convex disks with positive curvature, linking boundary length to proximity to spherical caps, and proves a compactness result for a related PDE problem.

Findings

Convex disks with large boundary length are close to spherical caps in Gromov-Hausdorff sense.

Proves a stability version of a theorem by Hang and Wang.

Provides a compactness result for a Liouville-type PDE.

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.