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The cross curvature flow of 3-manifolds with negative sectional curvature

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. | Turk J Math 28 (2004) , 1 – 10. ¨ ITAK ˙ c TUB The Cross Curvature Flow of 3-Manifolds with Negative Sectional Curvature Bennett Chow∗ , Richard S. Hamilton Abstract We consider the cross curvature flow, an evolution equation of metrics on 3manifolds. 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. 1. The evolution equation When n = 3, it is an old conjecture, which is also a consequence of the Geometrization Conjecture, that any closed 3-manifold with negative sectional curvature admits a hyperbolic metric. In this article, we introduce an evolution equation which deforms metrics on 3-manifolds with sectional curvature of one sign. Given a closed 3-manifold with an initial metric with negative sectional curvature, we conjecture that this flow will exist for all time and converge to a hyperbolic metric after a normalization. We shall establish some results, including monotonicity formulae, in support of this conjecture. Note that in contrast to negative sectional curvature, every closed n -manifold admits a metric with negative Ricci curvature by the work of Gao and Yau [6], [7] for n = 3 and Lohkamp [12] for all n ≥ 3. When n ≥ 4, Gromov and Thurston [8] have shown that there exist closed manifolds with arbitrarily pinched negative sectional curvature which do not admit metrics with constant negative sectional curvature. It is unknown whether such manifolds admit Einstein metrics. In particular, the stability result for Ricci flow of Ye [15] assumes more than just curvature pinching depending only on dimension.1 Let (M, g) be a 3-dimensional Riemannian manifold with negative sectional curvature. The .