The Collapsing Rate of the K\"ahler-Ricci Flow with Regular Infinite   Time Singularity
  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. 
Frederick Tsz-Ho Fong, Zhou Zhang
math.DG math.CV