Geometric inflexibility of hyperbolic cone-manifolds

TL;DR

This paper proves that 3D hyperbolic cone-manifolds exhibit geometric inflexibility, with bi-Lipschitz maps whose distortion decays exponentially with distance from the cone singularity, providing key estimates for deformation control.

Contribution

It establishes exponential decay estimates for bi-Lipschitz constants in hyperbolic cone-manifold deformations, advancing understanding of their geometric rigidity.

Findings

Bi-Lipschitz diffeomorphism with exponential decay in distortion

Estimates controlled by complex lengths of short curves

Enhanced understanding of geometric inflexibility in hyperbolic cone-manifolds

Abstract

We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lipschitz constant decays exponentially in the distance from the cone-singularity. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves.