Structure theory of non-collapsed limits of Ricci flows

Abstract

In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $\ge 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension $\ge 4$. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds. As an application of our theory, we obtain a description of the singularity formation of a Ricci flow at its first singular time and a thick-thin decomposition characterizing the long-time behavior of immortal flows. These results generalize Perelman's results in dimension 3 to higher dimensions. We also obtain a Backwards Pseudolocality Theorem and discuss several other applications.

155 pages; Comments and questions welcome; improved introduction and minor edits

References in corpus (2)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Ancient Ricci flows with asymptotic solitons

Cited by in corpus (5)

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• On a dichotomy of the curvature decay of steady Ricci soliton
• Ricci flow with bounded curvature integrals
• On the improved no-local-collapsing theorem of Ricci flow
• Type of finite time singularities of the Ricci flow with bounded scalar curvature