Characterization of Lipschitz Normally Embedded complex curves

Abstract

The main result of the paper states that a connected complex affine algebraic curve is Lipschitz normally embedded (shortened to LNE afterwards) in $\mathbb{C}^n$ if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in $\mathbb{CP}^n$ with the property of its general affine traces being LNE in $\mathbb{C}^n$. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.