Stability of the conical Kahler-Ricci flows on Fano manifolds

In this paper, we study stability of the conical Kahler-Ricci flows on Fano manifolds. That is, if there exists a conical Kahler-Einstein metric with cone angle 2 pi beta along the divisor, then for any beta' sufficiently close to beta, the corresponding conical Kahler-Ricci flow converges to a conical Kahler-Einstein metric with cone angle 2 pi beta' 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 Kahler-Einstein metrics with positive Ricci curvatures.