Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

TL;DR

This paper proves that certain 3D Ricci flows with nonnegative Ricci curvature become complete for positive times under specific initial conditions, extending previous results using pseudolocality techniques.

Contribution

It establishes conditions under which singular Ricci flows become complete, combining pseudolocality results and adapting previous constructions for nonnegative curvature.

Findings

Flow $g(t)$ is complete for positive times under volume ratio bounds.

Completeness is achieved for initial data close to nonnegative sectional curvature.

Method extends to flows with compactly supported perturbations of Euclidean space.

Abstract

Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3, g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on $\mathbb{R}^3$.