Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

TL;DR

This paper extends rigidity estimates for nearly umbilical surfaces from the 3D case to arbitrary codimension, using varifold monotonicity instead of classical Codazzi equations, providing new quantitative geometric stability results.

Contribution

It introduces a novel approach employing varifold monotonicity to establish rigidity estimates for nearly umbilical surfaces in any codimension, broadening previous 3D results.

Findings

Extended rigidity estimates to arbitrary codimension

Used varifold monotonicity instead of Codazzi equations

Provided quantitative stability results for nearly umbilical surfaces

Abstract

In [dLMu05], DeLellis and M\"uller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ \Sigma \subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(\Sigma) = 4 \pi\ $, it was shown that $\ \parallel A_\Sigma - id \parallel_{L^2(\Sigma)} \leq C \parallel A^0_\Sigma \parallel_{L^2(\Sigma)}\ $, and building on this, closeness of $\ \Sigma\ $ to a round sphere in $\ W^{2,2}\ $ was established, when $\ \parallel A^0_\Sigma \parallel_{L^2(\Sigma)}\ $ is small. This was supplemented in [dLMu06] by giving a conformal parametrization $\ S^2 \stackrel{\approx}{\longrightarrow} \Sigma\ $ with small conformal factor in $\ L^\infty\ $, again when $\ \parallel A^0_\Sigma \parallel_{L^2(\Sigma)}\ $ is small. In this article, we extend these results to arbitrary codimension. In contrast to [dLMu05], our argument is not based on the equation of Mainardi-Codazzi, but instead uses the monotonicity formula for varifolds.