Concentration of curvature and Lipschitz invariants of holomorphic functions of two variables

TL;DR

This paper demonstrates that Lipschitz homeomorphisms preserve multi-scale curvature concentration zones and gradient canyon structures in holomorphic functions of two variables, introducing new invariants in the field.

Contribution

It introduces the first new Lipschitz invariants since 2003 by linking curvature concentration with geometric structures in holomorphic functions.

Findings

Lipschitz homeomorphisms preserve zones of multi-scale curvature concentration

Gradient canyon structures are invariant under Lipschitz transformations

New invariants extend the understanding of geometric properties of holomorphic functions

Abstract

By combining analytic and geometric viewpoints on the concentration of the curvature of the Milnor fibre, we prove that Lipschitz homeomorphisms preserve the zones of multi-scale curvature concentration as well as the gradient canyon structure of holomorphic functions of two variables. This yields the first new Lipschitz invariants after those discovered by Henry and Parusinski in 2003.