The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2\pi$

TL;DR

This paper develops the deformation theory for hyperbolic cone-3-manifolds with cone-angles less than 2π, proving local rigidity and addressing a question posed by Casson.

Contribution

It establishes local rigidity for hyperbolic cone-3-manifolds with cone-angles in (0, 2π), advancing understanding of their deformation space.

Findings

Proves local rigidity for cone-3-manifolds with cone-angles less than 2π.

Provides a positive answer to Casson's question on deformation.

Develops foundational deformation theory for these structures.

Abstract

We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2\pi$, i.e. contained in the interval $(0,2\pi)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a question of A. Casson.