The Kähler–Ricci flow on surfaces of positive Kodaira dimension

Dimensional reduction and the long-time behavior of Ricci flow

“…This rescaling limit can be considered to give a canonical geometry for M . A similar phenomenon occurs in the work of Song and Tian concerning collapsing in the Kähler-Ricci flow on elliptic fibrations [52].…”

Gromov–Hausdorff collapsing of Calabi–Yau manifolds

“…Let α 0 be an ample class on N , and ω t ∈ f * α 0 + tα be the Ricci-flat Kähler metric given by Yau's Theorem [40] for t ∈ (0, 1], which satisfies the complex Monge-Ampère equation 2 Ω ∧ Ω. By [35] and [12],ω t converges smoothly to f * ω on f −1 (K) for any compact K ⊂ N 0 as t → 0, and on N 0 , where ω is the Kähler metric on N 0 with Ric(ω) = ω W P obtained in [35] and [29] (see also [28]), and ω W P is a Weil-Petersson semipositive form on N 0 coming from the variation of the complex structures of the fibers M y . Furthermore, the Ricci-flat metricsω t have locally uniformly bounded curvature on f −1 (N 0 ).…”

On the weak Kähler-Ricci flow

“…On a general Kähler manifold, the Kähler-Ricci flow (1.1) may develop singularity in finite time (see [31]). In an on-going project with his collaborators, the second named author proposed some problems of studying how the Kähler-Ricci flow extends across the finite time singularity (see [28] for more discussion). One of the problems involves constructing solutions of the Kähler-Ricci flow with much weaker initial metrics, possibly on spaces with mild singularity.…”