The Collapsing Rate of the Kähler-Ricci Flow with Regular Infinite Time Singularity
arXiv:1202.3199 · doi:10.1515/crelle-2013-0043
Abstract
We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with possibly singular fibers will also be discussed in local sense.
18 pages; final version, to appear in J. Reine Angew. Math
References in corpus (3)
Cited by in corpus (19)
- The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
- Remarks on the collapsing of torus fibered Calabi-Yau manifolds
- Infinite time singularities of the Kähler-Ricci flow
- Inoue surfaces and the Chern-Ricci flow
- The Anomaly flow over Riemann surfaces
- Higher-order estimates for collapsing Calabi-Yau metrics
- The Chern-Ricci flow on Oeljeklaus-Toma manifolds
- Bounding diameter of singular Kähler metric
- Local curvature estimates of long-time solutions to the Kähler-Ricci flow
- The Chern-Ricci flow
- Geometry of twisted Kähler-Einstein metrics and collapsing
- Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow
- A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry
- Twisted Kähler-Einstein Metrics and Collapsing
- Leafwise flat forms on Inoue-Bombieri surfaces
- Collapsing immortal Kähler-Ricci flows
- The Kähler-Ricci flow on compact Kähler manifolds
- On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows
- Integral inequalities for holomorphic maps and applications