On Lipschitz Rigidity of Complex Analytic Sets

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, any complex algebraic set in \(\mathbb {C}^n\) which is Lipschitz regular at infinity is an affine linear subspace.