# On Lipschitz rigidity of complex analytic sets

## TL;DR

This paper proves that complex analytic sets in complex Euclidean space which are Lipschitz normally embedded at infinity and have a linear tangent cone at infinity must be affine linear subspaces, revealing a rigidity property.

## Contribution

It establishes a Lipschitz rigidity result for complex analytic sets at infinity without restrictions on singularities, dimensions, or codimensions.

## Key findings

- Lipschitz normally embedded sets with linear tangent cones are affine linear.
- Lipschitz regular algebraic sets at infinity are affine linear.
- No restrictions on singularities or dimensions are needed for the rigidity.

## Abstract

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, a complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.