A note on the Ricci flow on noncompact manifolds

TL;DR

This paper proves the existence of long-time solutions to the Ricci flow on certain noncompact 3-manifolds with nonnegative Ricci curvature and scalar curvature tending to zero at infinity, extending recent results and including higher-dimensional and Kähler cases.

Contribution

It extends the existence results of Ricci flow to broader classes of noncompact manifolds, including higher dimensions and Kähler manifolds, under specific curvature conditions.

Findings

Long-time Ricci flow solutions on noncompact 3-manifolds with nonnegative Ricci curvature.

Extension of results to higher-dimensional manifolds.

Reproof of Kähler case analogous to previous work.

Abstract

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with initial data $(M^3,g_0)$ has a long time solution. This extends a recent result of Ma and Zhu. We also have a higher dimensional version, and we reprove a K$\ddot{a}$hler analogy due to Chau, Tam and Yu.