Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows (Q6581842)

The paper is about a conjecture of \textit{A. D. McLeod} and \textit{P. M. Topping} [Math. Z. 296, No. 1--2, 511--523 (2020; Zbl 1459.53086)]: If \((M^3,g_0)\) is a complete noncompact \(3\)-dimensional Riemannian manifold with nonnegative Ricci curvature, then the Ricci flow admits a smooth short time solution \(g(t)\) on \(M^3\times [0,T)\) for some \(T>0\), and \(g(t)\) is complete and has nonnegative Ricci curvature for each \(t\in [0,T)\).\N\NA recent progress towards this conjecture was obtained by \textit{Y. Lai} [Geom. Topol. 25, No. 7, 3629--3690 (2021; Zbl 1494.53104)], where it was proved if the condition of completeness of the solution \(g(t)\) is removed. Such a solution was constructed using the method of singular Ricci flows. In the paper under review, the authors show that the solution \(g(t)\) constructed by Y. Lai [loc. cit.] is complete provided that the initial metric \(g_0\) satisfies a volume ratio lower bound that approaches zero at spatial infinity.\N\NA key in the proof is to combine the pseudolocality theorem of Y. Lai [loc. cit.] for singular Ricci flows and a pseudolocality theorem of \textit{M. Simon} and \textit{P. M. Topping} [J. Differ. Geom. 122, No. 3, 467--518 (2022; Zbl 1529.53088)] for nonsingular Ricci flows to obtain a modification of Perelman's second pseudolocality theorem for singular Ricci flows.\N\NThe authors also show that if the initial volume ratio lower bound is replaced by the assumption that \(g_0\) is a compactly supported perturbation of a complete nonnegative sectional curvature metric, then the Ricci flow also admits a smooth short time solution \(g(t)\) which is complete and has nonnegative Ricci curvature for each \(t\in [0,T)\). The idea is to combine the result of \textit{E. Cabezas-Rivas} and \textit{B. Wilking} [J. Eur. Math. Soc. (JEMS) 17, No. 12, 3153--3194 (2015; Zbl 1351.53078)] and the construction of Y. Lai [loc. cit.].