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.

摘要

在本文中，我们研究了Fano流形上圆锥形Kähler-Ricci流的稳定性。也就是说，如果存在具有锥角的圆锥形Kähler-Einstein度量$2\pi \beta$ 沿着除数，那么对于任何 $\beta \prime$足够接近β时，相应的锥形Kähler-Ricci流收敛为锥角的锥形Kähler-Einstein度量$2\pi \beta \prime$沿除数。在此，我们仅使用Log Mabuchi能量从下方限制的条件。这比我们研究收敛性的适当性要弱。作为应用程序，我们给出了唐纳森开放性定理及其对存在正Ricci曲率的圆锥形Kähler-Einstein度量的猜想的抛物线证明。