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

We prove that the conical Kähler-Ricci flows introduced in [11] exist for all time t € .0;C1/. These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C₁,β 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 ℂn) with conical singularities.