Compactness and partial regularity theory of Ricci flows in higher dimensions

Théorie de la compacité et de la régularité partielle des écoulements de Ricci en dimensions supérieures

Richard Bamler (1) , Fanny Bastien (2) , Hugo Béchet (2)

1 UC Berkeley - University of California [Berkeley]
2 IF - Institut Fourier

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending 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 we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.

Mots clés: eem2021; grenoble; curvature constraints and spaces of metrics; Ricci flows; higher dimensions; eem2021

Mots clés: contraintes de courbures et espaces métriques; contraintes de courbures et espaces métriques

Domaines: Mathématiques [math]

Richard Bamler, Fanny Bastien, Hugo Béchet. Compactness and partial regularity theory of Ricci flows in higher dimensions: Curvature Constraints and Spaces of Metrics. 2021. ⟨hal-03677322⟩