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

The classic 2π–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 “2π–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.

cross curvature flow, 2$\pi$–theorem
Mathematical Subject Classification 2000
Primary: 53C44
Secondary: 57M50, 58J35, 58J32
Received: 24 June 2009
Revised: 25 November 2009
Accepted: 17 December 2009
Published: 1 March 2010
Jason DeBlois
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
322 Science and Engineering Offices (M/C 249)
851 S Morgan Street
Chicago, IL 60607-7045
Dan Knopf
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin, TX 78712-0257
Andrea Young
Department of Mathematics
University of Arizona
617 N Santa Rita Ave
PO Box 210089
Tucson, AZ 85721-0089