- Turkish Journal of Mathematics
- Vol: 28 Issue: 1
- The Cross Curvature Flow of 3-Manifolds with Negative Sectional Curvature
The Cross Curvature Flow of 3-Manifolds with Negative Sectional Curvature
Authors : Bennett Chow, Richard S. Hamilton
Pages : 1-10
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Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :We consider the cross curvature flow, an evolution equation of metrics on 3-manifolds. We establish short time existence when the sectional curvature has a sign. In the case of negative sectional curvature, we obtain some monotonicity formulas which support the conjecture that after normalization, for initial metrics on closed 3-manifolds with negative sectional curvature, the solution exists for all time and converges to a hyperbolic metric. This conjecture is still open at the present time.Keywords : Cross curvature flow, geometric evolution equation, negative sectional curvature, monotonicity formula, hyperbolic metric