Ricci Flow and Nonnegativity of Curvature

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four 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. The example is the first of this type. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow. §0 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 the curvature. In dimension three, Hamilton [ H1] proves that on compact manifolds the Ricc 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 admits 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 high dimension 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 manifold 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. Research partially supported by NSF grant DMS-0203023, USA.