The Cross Curvature Flow of 3-manifolds with Negative Sectional Curvature

2003/08/31 by Bennett Chow, Richard S. Hamilton, Chow, Bennett +2
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG #math.GT #msc:53C44

paper · pdf · doi:10.48550/arxiv.math/0309008

6 pages, submitted to the Proceedings of the 10th Gokova Geometry Topology Conference, May 26-31, 2003, Gokova, Turkey

arxiv created 2003/08/31 · arxiv updated 2009/12/01

Abstract

We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas shows that, provided the solution exists for all time, the metric approaches hyperbolic in an integral sense. Long time existence is still an open problem.

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