Asymptotic Stability of the Cross Curvature Flow at a Hyperbolic Metric

We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and converges to a hyperbolic metric. We show that the same technique proves an analogous result for Ricci flow. Additionally, we show short time existence and uniqueness of cross curvature flow for a more general class of initial data than was previously known. It has been conjectured that any 3-manifold with negative sectional curvature admits a hyperbolic metric; this conjecture follows from Thurston’s geometrization conjecture. However, it is unclear whether Ricci flow would provide a direct proof of the hyperbolic metric conjecture, as one expects hyperbolic pieces only to appear at large times and after perhaps multiple surgeries. Additionally, Ricci flow does not preserve negative curvature in general; so it would be useful to have an alternative flow. In 2004, Richard Hamilton and Bennett Chow proposed the cross curvature flow [2] and conjectured that in dimension 3 it would in fact preserve negative sectional curvature. They further conjectured that, given a metric g0 having negative sectional curvature, one could use a suitably normalized cross curvature flow to find a 1-parameter family of metrics g(t) having negative sectional curvature that converge to a hyperbolic metric as t approaches infinity. Notice that this would allow the space of hyperbolic metrics to be exhibited as a deformation retract of the space of metrics of negative sectional curvature in dimension three. This is certainly not the case in higher dimensions. For example, F. Thomas Farrell and Pedro Ontaneda show that, in dimensions n ≥ 10, the space of negatively curved metrics on a compact manifold M that admits a metric of strictly negative sectional curvature has infinitely many path components [4]. In this paper, we show that the cross curvature flow is asymptotically stable at a hyperbolic metric, thus providing new evidence that cross curvature flow may be fruitful in the pursuit of the above conjectures. In the appendix, we also apply the methods developed in this paper to give a new, simple proof of stability of Ricci flow at hyperbolic metrics in dimension three. (The dynamic stability of Ricci flow starting at negatively-curved metrics satisfying certain pinching hypotheses and other geometric bounds was studied by Rugang Ye in 1993 using alternate methods. His result is a priori stronger, since it does not assume existence of a hyperbolic metric [10].) The cross curvature flow (XCF) is a fully non-linear, weakly parabolic system of equations, which can be defined as follows: let Pab = Rab − 1 2Rgab be the Einstein tensor, and let P ij = ggPab. We can define the cross curvature tensor, Xij , to First author partially supported by NSF grants DMS-0505920 and DMS-0545984. 1 2 DAN KNOPF AND ANDREA YOUNG