Almost splitting and quantitative stratification for super Ricci flow
Keita Kunikawa, Yohei Sakurai
TL;DR
This work extends Bamler’s regularity theory for Ricci flow to the broader setting of super Ricci flows with non-negative Müller quantity \\( \mathcal{D}\ge 0 \\). The authors develop a framework based on heat kernels, Nash entropy, and Wasserstein geometry to obtain almost rigidity results: an almost static cone-splitting theorem, an almost splitting theorem, and a quantitative stratification theorem for \\( \\mathbb{F} \\\text{-limits} \\) of SRFs. A key byproduct is an almost-constancy statement for a scalar-curvature integral at almost selfsimilar points, which is new even in the classical Ricci-flow setting. The approach hinges on monotonicity and derivative estimates for Nash entropy, robust heat-kernel bounds, and parabolic rescaling arguments that yield controlled geometric and measure-theoretic behavior on small scales, enabling a stratified regularity theory in the non-smooth regime. These results provide a quantitative dimensionally robust picture of singularity models and regularity in high-dimensional SRFs, with potential implications for convergence theory and metric-measure geometry under lower curvature bounds.
Abstract
The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. As a byproduct, we obtain an almost constancy for a certain integral quantity concerning scalar curvature at an almost selfsimilar point, which is new even for Ricci flow.