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

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

2003-08-31

arxiv.org/abs/math/0309008v1

Mathematics

Differential Geometry

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

Scientific paper

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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