On the long time behaviour of the Conical Kähler- Ricci flows

Abstract

We prove that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time $t\in [0,+\infty )$. These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class $C_{1,\beta }$ is negative or zero, the corresponding conical Kähler-Ricci flows converge to Kähler-Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville type theorem for Kähler-Ricci flat metrics (which are defined over ${\mathbb{C}}^n$) with conical singularities.

50 pages. Comments are welcome

References in corpus (9)

• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Ricci flow on surfaces with conic singularities
• The Kahler-Ricci flow through singularities
• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Conical Kahler-Einstein metric revisited
• Smooth approximations of the Conical Kahler-Ricci flows
• The conical Kähler-Ricci flow on Fano manifolds
• Existence of Weak Conical Kähler-Einstein Metrics Along Smooth Hypersurfaces

Cited by in corpus (10)

• The conical Kähler-Ricci flow on Fano manifolds
• On the existence of constant scalar curvature Kähler metric: a new perspective
• Unnormalize conical Kähler-Ricci flow
• On the regularity problem of complex Monge-Ampere equations with conical singularities
• $C^{2,\alpha }$-estimate for conical Kähler-Ricci flow
• The conical Kähler-Ricci flow with weak initial data on Fano manifold
• 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
• Metric contraction of the cone divisor by the conical Kähler-Ricci flow