A characterization of Lipschitz normally embedded surface singularities

TL;DR

This paper provides a precise criterion for when a normal surface singularity has bilipschitz equivalent outer and inner metrics, characterizing Lipschitz normally embedded singularities in complex analytic spaces.

Contribution

It establishes a necessary and sufficient condition for Lipschitz normal embedding of normal surface singularities, advancing understanding of metric equivalences in complex geometry.

Findings

Rational surface singularities are LNE if and only if they are minimal.

Provides a characterization criterion for Lipschitz normally embedded surface singularities.

Connects metric properties with algebraic classification of surface singularities.

Abstract

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), i.e., to have bilipschitz equivalent outer and inner metrics. In a partner paper [15] we apply it to prove that rational surface singularities are LNE if and only if they are minimal.