The bi-Lipschitz constant of an isothermal coordinate chart

Abstract

Let $M$ be a $C^{2}$-smooth Riemannian surface. A classical theorem in differential geometry states that the Gauss curvature function $K : M \to \mathbb{R}$ vanishes everywhere if and only if the surface is locally isometric to the Euclidean plane. We give an asymptotically sharp quantitative version of this theorem with respect to an isothermal coordinate chart. Roughly speaking, we show that if $B$ is a Riemannian disc of radius $\delta > 0$ with $\delta^{2}\sup_{B}|K| < \varepsilon$ for some $0 < \varepsilon < 1$, then there is an isothermal coordinate map from $B$ onto an Euclidean disc of radius $\delta$ which is bi-Lipschitz with constant $\exp(4 \varepsilon)$.