Ricci Flow and Nonnegativity of Sectional Curvature

In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional curvature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow. Introduction The Ricci flow has been proved to be an effective tool in the study of the geometry and topology of manifolds. One of the good properties of the Ricci flow is that it preserves the ‘nonnegativity’ of various curvatures. In dimension three, Hamilton [H1] proves that on compact manifolds the Ricci flow preserves the nonnegativity of the Ricci curvature and the sectional curvature. Using this property and the quantified version, curvature pinching estimate, it was proved in [H1] that the normalized Ricci flow converges to a Einstein metric if the initial metric has positive Ricci curvature. In particular, it implies that a simply-connected compact three-manifold is diffeomorphic to the three sphere if it admits a metric with positive Ricci curvature. One can refer [Ch] for an updated survey and [P2] for some recent developement on the Ricci flow on three manifolds. Later in [H2] it was proved that the Ricci flow also preserves the nonnegativity of the curvature operator in all dimensions (on compact manifolds). In the Kähler case, Bando and Mok [B, M] proved that the flow also preserves the nonnegativity of the holomorphic bisectional curvature. The Ricci flow on complete manifolds was initiated in [Sh2]. In [Sh3] Shi generalized the above mentioned result of Bando and Mok to the complete Kähler manifolds with bounded curvature. Interesting applications were also obtained therein. In this paper, we shall study the topological consequences of the assumption that Ricci flow preserves the nonnegativity of the sectional curvature on complete Riemannian manifolds. The basic method is to study the heat equation, time Received May 3, 2003. Revised August 2004. Research partially supported by NSF grant DMS-0328624, USA.