Lipschitz geometry of the image of finite mappings

TL;DR

This paper investigates the Lipschitz normally embedded (LNE) property of images of finite complex analytic map germs, establishing conditions under which the image is smooth or an embedding based on multiplicity and injectivity.

Contribution

It characterizes when the image of finite map germs is LNE, linking smoothness and embeddings to multiplicity and injectivity conditions.

Findings

Image is LNE iff it is smooth when multiplicity equals generic degree.

Finite corank 1 maps satisfy the LNE condition.

Injective maps have LNE images iff they are embeddings.

Abstract

This paper is devoted to the study of the LNE property in complex analytic hypersurface parametrized germs, that is, the sets that are images of finite analytic map germs from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$. We prove that if the multiplicity of $f$ is equal to his generic degree, then the image of $f$ is LNE at 0 if and only if it is a smooth germ. We also show that every finite corank 1 map is sattisfies the previous hypothesis. Moreover, we show that for an injective map germ $f$ from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$, the image of $f$ is LNE at 0 if and only if $f$ is an embedding.