Volume 10, issue 1 (2010)

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Cross curvature flow on a negatively curved solid torus

Jason DeBlois, Dan Knopf and Andrea Young

Algebraic & Geometric Topology 10 (2010) 343–372
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.

Keywords
cross curvature flow, 2$\pi$–theorem
Mathematical Subject Classification 2000
Primary: 53C44
Secondary: 57M50, 58J35, 58J32