Minimal surface singularities are Lipschitz normally embedded

TL;DR

This paper proves that minimal surface singularities in complex analytic spaces are Lipschitz normally embedded, meaning their outer and inner metrics are bilipschitz equivalent, and they are uniquely characterized among rational surface singularities.

Contribution

It establishes that minimal surface singularities are Lipschitz normally embedded and uniquely identifies them among rational surface singularities based on this property.

Findings

Minimal surface singularities are Lipschitz normally embedded.

They are the only rational surface singularities with this property.

Outer and inner metrics are bilipschitz equivalent for these singularities.

Abstract

Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property.