Deformations of Lipschitz Homeomorphisms

TL;DR

This paper develops Lipschitz analogues of key topological deformation results, including Perelman's gluing theorem, with an emphasis on tracking Lipschitz constants, advancing the understanding of Lipschitz homeomorphisms.

Contribution

It introduces Lipschitz deformation theories and gluing theorems, extending classical topological results to the Lipschitz setting with constant control.

Findings

Lipschitz deformation results analogous to Siebenmann's theory

Lipschitz gluing theorem established

Deformation theory with Lipschitz constant tracking developed

Abstract

We obtain the Lipschitz analogues of the results Perelman used from Siebenmann's deformation of homeomorphism theory in his proof of the stability theorem. Consequently, we obtain the Lipschitz analogue of Perelman's gluing theorem. Moreover, we obtain the analogous deformation theory but with tracking of the Lipschitz constants.