Stability of the conical K\"ahler-Ricci flows on Fano manifolds
Abstract
In this paper, we study the stability of the conical K\"ahler-Ricci flows on Fano manifolds. That is, if there exists a conical K\"ahler-Einstein metric with cone angle 2πβ along the divisor, then for any β' sufficiently close to β, the corresponding conical K\"ahler-Ricci flow converges to a conical K\"ahler-Einstein metric with cone angle 2πβ' 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 before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical K\"ahler-Einstein metrics with positive Ricci curvatures.
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