How to produce a Ricci Flow via Cheeger-Gromoll exhaustion

TL;DR

This paper establishes short and long-term existence results for Ricci flow on open manifolds with nonnegative complex sectional curvature, using Cheeger-Gromoll exhaustion and constructing explicit solutions with unbounded curvature.

Contribution

It introduces a novel method to produce Ricci flows on open manifolds without upper curvature bounds using convex exhaustion techniques.

Findings

Proves short time existence of Ricci flow on noncompact manifolds.

Identifies volume growth conditions for long-term existence.

Constructs an explicit immortal solution with unbounded curvature.

Abstract

We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with nonnegative complex sectional curvature which subconverge to a solution of the Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time existence, and we give an analysis of the long time behaviour of the Ricci flow. Finally, we construct an explicit example of an immortal nonnegatively curved solution of the Ricci flow with unbounded curvature for all time.