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The bi-Lipschitz constant of an isothermal coordinate chart

Matan Eilat

TL;DR

The paper delivers a sharp quantitative refinement of the flatness theorem for $2$-D Riemannian surfaces by exploiting isothermal coordinates. It constructs harmonic barrier functions based on Green’s function and CAT$(\kappa)$ geometry to bound the conformal factor $\varphi$ of the isothermal chart, connecting curvature, radius, and the Liouville equation to explicit distance distortion bounds. The main result provides an explicit bound on $\sup_B|\log\varphi|$ in terms of $\delta^{2}\kappa$ and a hyperbolic-trigonometric factor, yielding bi-Lipschitz distortion bounds $\exp(\pm 4\delta^{2}\kappa)$ under a stronger smallness condition. The approach is complemented by detailed boundary analysis and barrier inequalities, with an appendix of technical lemmas supporting the estimates. Overall, the work offers asymptotically sharp, practical criteria for near-flat metrics in isothermal coordinates, enabling controlled local geometric injections into the Euclidean plane.

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 $δ> 0$ with $δ^{2}\sup_{B}|K| < \varepsilon$ for some $0 < \varepsilon < 1$, then there is an isothermal coordinate map from $B$ onto an Euclidean disc of radius $δ$ which is bi-Lipschitz with constant $\exp(4 \varepsilon)$.

The bi-Lipschitz constant of an isothermal coordinate chart

TL;DR

The paper delivers a sharp quantitative refinement of the flatness theorem for -D Riemannian surfaces by exploiting isothermal coordinates. It constructs harmonic barrier functions based on Green’s function and CAT geometry to bound the conformal factor of the isothermal chart, connecting curvature, radius, and the Liouville equation to explicit distance distortion bounds. The main result provides an explicit bound on in terms of and a hyperbolic-trigonometric factor, yielding bi-Lipschitz distortion bounds under a stronger smallness condition. The approach is complemented by detailed boundary analysis and barrier inequalities, with an appendix of technical lemmas supporting the estimates. Overall, the work offers asymptotically sharp, practical criteria for near-flat metrics in isothermal coordinates, enabling controlled local geometric injections into the Euclidean plane.

Abstract

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

Paper Structure

This paper contains 5 sections, 14 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The harmonic barriers
  4. The ratio between distances
  5. Appendix

Key Result

Theorem 1.1

Let $M$ be a $C^{2}$-smooth Riemannian surface, fix $p_{0} \in M$ and let $\delta > 0$. Suppose that at any point $p \in B = B_{M}(p_{0},\delta)$ the injectivity radius is at least $2\delta$ and $-\kappa \leq K(p) \leq \kappa$, where $\kappa > 0$ is a constant satisfying $\delta^{2}\kappa < \pi^{2}

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • ...and 16 more