Stability of the conical Kähler-Ricci flows on Fano manifolds

Abstract

In this paper, we study stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2\pi \beta$ along the divisor, then for any $\beta \prime$ sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle $2\pi \beta \prime$ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence. As applications, we give parabolic proofs of Donaldson’s openness theorem and his conjecture for the existence of conical Kähler-Einstein metrics with positive Ricci curvatures.

Keywords:

• Stability
• conical Kähler-Einstein metric
• conical Kähler-Ricci flow
• twisted Kähler-Ricci flow

2010 Mathematics Subject Classification:

• 53C44
• 35K96

Acknowledge

The first author would like to thank Professors Jiayu Li, Miles Simon and Xiaohua Zhu for their useful discussions, consistent help and support. He also would like to thank Professor Xiangwen Zhang, Doctors Xishen Jin and Chao Li for their helpful discussions. Part of this work was carried out while the first author’s visit to the University of Newcastle and the University of Adelaide in Australia. He is grateful to Professors James McCoy and Thomas Leistner for their invitations, and the universities for their hospitality. The authors would love to express their gratitude toward the referees for their valuable comments.

Additional information

Funding

The first author is supported by the Special Priority Program SPP 2026 “Geometry at Infinity” from the German Research Foundation (DFG). The second author is supported by NSF in China No.11625106, 11571332 and 11721101.