Bi-Lipschitz rigidity for $L^2$-almost CMC surfaces

TL;DR

This paper establishes optimal bi-Lipschitz and $W^{2,2}$ parametrization estimates for surfaces in the unit ball with near-constant density and small Willmore energy, and explores quantitative rigidity for $L^2$-almost CMC surfaces.

Contribution

It provides the first optimal a priori estimates and rigidity results for $L^2$-almost constant mean curvature surfaces in the unit ball.

Findings

Optimal bi-Lipschitz and $W^{2,2}$ estimates for surfaces with small Willmore energy.

Quantitative rigidity results for $L^2$-almost CMC surfaces.

Extension of rigidity theory to surfaces with near-constant density.

Abstract

For smooth surfaces properly immersed in the unit ball of $\RR^n$ with density close to one and small Willmore energy, the optimal a priori estimate(bi-Lipschitz and $W^{2,2}$ parametrization)is provided. We also discuss the quantitative rigidity for $L^2$-almost CMC surfaces.