Nonsingular Ricci flow on a noncompact manifold in dimension three

TL;DR

This paper proves that the Ricci flow on a 3D complete noncompact manifold with non-negative curvature operator remains nonsingular for any finite time, advancing understanding of geometric evolution in noncompact settings.

Contribution

It establishes the nonsingularity of Ricci flow on noncompact 3D manifolds with non-negative curvature operator, a result not previously confirmed.

Findings

Ricci flow remains nonsingular in finite time

Non-negative curvature operator ensures flow stability

Results apply to complete noncompact manifolds

Abstract

We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.