The bi-Lipschitz constant of an isothermal coordinate chart

TL;DR

This paper provides a quantitative estimate of how close a Riemannian surface with small curvature is to being flat, by analyzing the bi-Lipschitz constant of isothermal coordinate charts.

Contribution

It establishes an asymptotically sharp bound on the bi-Lipschitz constant for isothermal charts in terms of the curvature, refining classical results in differential geometry.

Findings

Bi-Lipschitz constant is bounded by exp(4ε) for small curvature.

The result is asymptotically sharp as curvature approaches zero.

Provides a quantitative version of the classical flatness theorem.

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 $\exp(4 \varepsilon)$.