Cross curvature flow on a negatively curved solid torus

Abstract

The classic $2\pi$–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 “$2\pi$–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.