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arXiv · 2307.08088

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

Albert Chau·Adam Martens
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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 $\mathbb{R}^3$.

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Albert Chau, Adam Martens. 2023-07-16. Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows. https://arxiv.org/abs/2307.08088

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