What question did this study set out to answer?

The aim is to provide a quantitative version of a classical theorem about the relationship between Gauss curvature and local isometry to the Euclidean plane.

February 22, 2026

The bi-Lipschitz constant of an isothermal coordinate chart

Key Points

The aim is to provide a quantitative version of a classical theorem about the relationship between Gauss curvature and local isometry to the Euclidean plane.
Analyzed Riemannian discs with specific curvature properties.
Developed an isothermal coordinate map.
Studied the bi-Lipschitz constant in relation to the coordinate mapping.
Demonstrated that the bi-Lipschitz constant relates to the curvature of the surface.
Provided an asymptotic bound for the constant in Riemannian discs.

Abstract

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.

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