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

Authors: BENNETT CHOW, RICHARD S. HAMILTON

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

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