Ricci flow and nonnegativity of curvature

TL;DR

This paper develops a maximum principle for the Ricci flow on symmetric tensors and constructs the first known example of a complete manifold with nonnegative sectional curvature in dimension ≥4 where Ricci flow fails to preserve this nonnegativity.

Contribution

It introduces a general maximum principle for the Lichnerowicz heat equation under Ricci flow and provides the first example of non-preservation of nonnegative sectional curvature in higher dimensions.

Findings

Constructed a complete manifold with bounded nonnegative sectional curvature in dimension ≥4.

Showed Ricci flow does not preserve nonnegativity of sectional curvature in higher dimensions.

Established a splitting theorem for metrics with nonnegative sectional curvature under Ricci flow.

Abstract

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.