Cross curvature flow on a negatively curved solid torus

DeBlois, Jason¹ ; Knopf, Dan² ; Young, Andrea³

¹Department 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, USA
²Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA
³Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, PO Box 210089, Tucson, AZ 85721-0089, USA

Algebraic and Geometric Topology, Tome 10 (2010) no. 1, pp. 343-372

Résumé

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.

Détail

DOI : 10.2140/agt.2010.10.343

Keywords: cross curvature flow, 2$\pi$–theorem

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