The conical Kähler-Ricci flow on Fano manifolds

Abstract

In this paper, we study the long-term behavior of the conical Kähler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted Kähler-Ricci flows, we obtain a long-time solution to the conical Kähler-Ricci flow by limiting a sequence of these twisted flows. Second, we study the uniform Perelman's estimates of the twisted Kähler-Ricci flows. After that, we prove that the conical Kähler-Ricci flow must converge to a conical Kähler-Einstein metric if there exists one.

46 pages

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• On the regularity problem of complex Monge-Ampere equations with conical singularities
• The conical Kähler-Ricci flow with weak initial data on Fano manifold
• Twisted and conical Kähler-Ricci soliton on Fano manifolds
• A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces
• $C^{2,\alpha }$-estimate for Monge-Ampere equations with Hölder-continuous right hand side
• Smoothing conic Kähler metrics with uniformly upper bisectional curvature bound
• Ricci measure for some singular Riemannian metrics
• Stability of the conical Kähler-Ricci flows on Fano manifolds