Global bi-Lipschitz classification of semi-algebraic surfaces

TL;DR

This paper provides a comprehensive bi-Lipschitz classification of semi-algebraic surfaces with isolated singularities, including Nash surfaces, complex algebraic curves, and minimal surfaces with finite total curvature, using inner distance metrics.

Contribution

It introduces a complete classification framework for semi-algebraic surfaces up to bi-Lipschitz homeomorphisms, extending to Nash surfaces and complex algebraic curves.

Findings

Complete classification for Nash surfaces

Classification of complex algebraic curves

Results on minimal surfaces with finite total curvature

Abstract

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature.