Optimal rigidity estimates for nearly umbilical surfaces

Abstract

Let $\Sigma \in \mathbf{R}^3$ be a smooth compact connected surface without boundary and denote by $A$ its second fundamental form. We prove the existence of a universal constant $C$ such that \begin{equation} \inf_{\lambda\in {\bf R}}\Vert A - \lambda \rm{Id} \Vert_{L^2(\Sigma)} \leq C \Vert A - \frac{\rm{tr}A}{2} \rm {Id} \Vert_{L^2(\Sigma)^\cdot} \end{equation} Building on this, we also show that, if the right-hand side of (1) is smaller than a geometric constant, Σ is W^2,2–close to a round sphere.