Gromov-Hausdorff Limits of Noncollapsed K\"ahler-Ricci Flows and the Geometry of Ricci Shrinkers

Abstract

We study the metric geometry of finite-time singularities of volume-noncollapsed K\"ahler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed K\"ahler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For K\"ahler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a K\"ahler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for K\"ahler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an $S^1$-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.