Cross curvature flow on a negatively curved solid torus

TL;DR

This paper explores using cross curvature flow to deform metrics on negatively curved solid tori from a 2pi-metric to a hyperbolic metric, providing partial progress on the deformation process.

Contribution

It introduces a program employing cross curvature flow to connect specific negatively curved metrics to hyperbolic metrics on 3-manifolds.

Findings

Proved long-time existence of the flow

Established preservation of negative sectional curvature

Demonstrated convergence to hyperbolic metric

Abstract

The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-metric" and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds, and integral convergence to hyperbolic for the metrics under consideration.