On Lipschitz Geometry at infinity of complex analytic sets

TL;DR

This paper proves that entire complex analytic sets with Lipschitz regularity at infinity are affine linear subspaces and characterizes algebraic sets via bi-Lipschitz homeomorphisms at infinity, extending classical Bernstein results.

Contribution

It establishes a complex non-parametric version of Moser's Bernstein Theorem and characterizes algebraic sets through bi-Lipschitz homeomorphisms at infinity.

Findings

Entire complex analytic sets Lipschitz regular at infinity are affine linear subspaces.

Bi-Lipschitz homeomorphism at infinity preserves algebraicity of complex analytic sets.

No restrictions on singularities, dimension, or codimension in the main results.

Abstract

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity must be an affine linear subspace of $\mathbb{C}^n$. In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if $X$ and $Y$ are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then $X$ is a complex algebraic set if and only if $Y$ is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here.