Uniqueness and stability of singular Ricci flows in higher dimensions

TL;DR

This paper extends the uniqueness and stability results of singular Ricci flows from three dimensions to higher dimensions under certain conditions, utilizing recent classifications and maximum principles.

Contribution

It generalizes the Bamler-Kleiner proof to higher dimensions, providing a canonical evolution through singularities for manifolds with positive isotropic curvature.

Findings

Uniqueness and stability of higher-dimensional singular Ricci flows established.

Applicable to manifolds with positive isotropic curvature.

Utilizes recent classification of higher-dimensional κ-solutions.

Abstract

In this short note, we observe that the Bamler-Kleiner proof of uniqueness and stability for 3-dimensional Ricci flow through singularities generalizes to singular Ricci flows in higher dimensions that satisfy an analogous canonical neighborhood property. In particular, this gives a canonical evolution through singularities for manifolds with positive isotropic curvature. The new ingredients we use are the recent classification of higher dimensional $\kappa$-solutions by Brendle, Daskalopoulos, Naff and Sesum, and the maximum principle for the linearized Ricci-DeTurck flow on locally conformally flat manifolds due to Chen and Wu.