On the long time behaviour of the conical Kähler–Ricci flows

Abstract

Abstract We prove that the conical Kähler–Ricci flows introduced in [11] exist for all time t ∈ [ 0 , + ∞ ) {t\in[0,+\infty)} . These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C 1 , β {C_{1,\beta}} is negative or zero, the corresponding conical Kähler–Ricci flows converge to Kähler–Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville-type theorem for Kähler–Ricci flat metrics (which are defined over ℂ n {\mathbb{C}^{n}} ) with conical singularities.