Stability of the conical Kähler-Ricci flows on Fano manifolds

Abstract

In this paper, we study the 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 $\beta$, 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 before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures.

51 pages, we improve the writing and add a remark

References in corpus (6)

• The logarithmic Sobolev inequality along the Ricci flow
• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• Uniform Sobolev inequalities for manifolds evolving by Ricci flow
• Schauder estimates for equations with cone metrics, I
• Perelman's $W$-functional on manifolds with conical singularities
• Perelman's Entropies for Manifolds with conical Singularities