Laplace comparison on Kähler Ricci flow and convergence
Gang Tian, Qi S. Zhang, Zhenlei Zhang, Meng Zhu, Xiaohua Zhu
TL;DR
The paper addresses the long-standing Hamilton-Tian convergence problem for the normalized Kähler-Ricci flow on Fano manifolds by proving a uniform integral Laplace comparison via a conformally modified metric $\tilde{g}=h g(t)$ with $h=e^{2f}$. The conformal approach uses $f$ solving a Schrödinger-type equation tied to the negative part of the Ricci curvature to control curvature terms and derive an $L^q$ bound for $\psi_+$, enabling a robust Laplace comparison despite lacking pointwise Ricci bounds. This framework extends Cheeger-Colding theory to the flow, yielding Gromov-Hausdorff convergence to a shrinking Kähler-Ricci soliton away from a codimension $4$ singular set, and demonstrates that the singular set in the limit has real codimension at least $4$ when pulled back to the original metric. The results unify and deepen existing convergence analyses by integrating conformal geometry with Kato-class Ricci control, and clarify connections to prior Bergman/kodaira approaches in the literature. Overall, the work provides a self-contained route to Hamilton-Tian-type convergence under weaker curvature hypotheses, with precise control of the singular set and metric structure on the regular part.
Abstract
We first prove a uniform integral Laplace comparison result for the Kähler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.