The bi-Lipschitz constant of an isothermal coordinate chart

Let [Formula: see text] be a [Formula: see text]-smooth Riemannian surface. A classical theorem in differential geometry states that the Gauss curvature function [Formula: see text] 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 [Formula: see text] is a Riemannian disc of radius [Formula: see text] with [Formula: see text] for some [Formula: see text], then there is an isothermal coordinate map from [Formula: see text] onto an Euclidean disc of radius [Formula: see text] which is bi-Lipschitz with constant [Formula: see text].