Deformations of Non-Compact, Projective Manifolds

TL;DR

This paper investigates the rigidity of hyperbolic structures on certain knots and links, showing some cannot be deformed into convex projective structures, while others admit non-trivial deformations under specific conditions.

Contribution

It establishes new rigidity results for hyperbolic structures on two-bridge knots and links and identifies conditions for non-trivial convex deformations of branched covers.

Findings

Complete hyperbolic structures on certain knots are rigid against convex projective deformations.

Some branched covers of amphicheiral knots admit non-trivial convex deformations.

Rigidity depends on specific geometric and topological hypotheses.

Abstract

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivial, strictly convex deformations near their complete hyperbolic structure.