Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

Abstract

De Lellis and Müller (J Differ Geom 69:75–110, 2005) proved a quantitative version of Codazzi’s theorem, namely for a smooth embedded surface ${\Sigma \subseteq \mathbb{R}^3 }$ with area normalized to ${\mathcal{H}^2(\Sigma) = 4 \pi}$, 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 De Lellis and Müller (Calc Var Partial Differ Equ 26(3):288–296, 2006) 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 De Lellis and Müller (J Differ Geom 69:75–110, 2005), our argument is not based on the equation of Mainardi–Codazzi, but instead uses the monotonicity formula for varifolds.