On algebraic bi-Lipschitz homeomorphisms

TL;DR

This paper investigates algebraic bi-Lipschitz homeomorphisms between complex varieties and germs, establishing invariance of degree and multiplicity but not normality under such mappings.

Contribution

It proves that degree and multiplicity are preserved under algebraic and c-holomorphic bi-Lipschitz homeomorphisms, respectively, and shows normality is not invariant.

Findings

Degree is preserved under algebraic bi-Lipschitz homeomorphisms.

Multiplicity is preserved under c-holomorphic bi-Lipschitz mappings.

Normality is not preserved under bi-Lipschitz homeomorphisms.

Abstract

Let $X\subset \mathbb{C}^n; Y\subset \mathbb{C}^m$ be closed affine varieties and let $\phi: X\to Y$ be an algebraic bi-Lipschitz homeomorphism. Then ${\rm deg}\ X={\rm deg}\ Y.$ Similarly, let $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset (\mathbb{C}^m,0)$ be germs of analytic sets and let $f: (X,0)\to (Y,0)$ be a c-holomorphic and bi-Lipschitz mapping. Then ${\rm mult}_0 \ X= {\rm mult }_0 \ Y.$ Finally we show that the normality is not a bi-Lipschitz invariant.