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

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 a monotonicity formula 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 [5] for n = 3 and Lohkamp [10] for all n ≥ 3. When n ≥ 4, Gromov and Thurston [6] 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 of Ye [13] assumes more than curvature pinching. 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. Let (M, g) be a 3-dimensional Riemannian manifold with negative sectional curvature. The Einstein tensor is Pij = Rij − 1 2Rgij . We find it convenient to raise the indices: P ij = ggRkl − 1 2Rg ij . The cross curvature tensor is