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arXiv · 2606.12379

A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature -- Part II

Reto Buzano·Gianmichele Di Matteo
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Abstract

We continue our local singularity analysis for Ricci flow initiated in ArXiv:2006.16227. Building on that framework, we study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound, and prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results of the first author with Enders and Topping and with Mantegazza that relied on a global Type I assumption. We then adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to negative infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order.

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Reto Buzano, Gianmichele Di Matteo. 2026-06-10. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature -- Part II. https://arxiv.org/abs/2606.12379

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