Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow

Abstract

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics, that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges in C ∞ loctopology away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.

On the Hölder estimate of Kähler-Ricci flow

“…) which remains uniformly equivalent to the Euclidean metric ω C m+n for all t ≤ 0. Now, we follow similar argument in [3] to reduce the discussion back to the elliptic case. Using (3.29), ω• ∞ (t) solves the Kähler-Ricci flow:…”

“…Applying the argument of [10,5.3.1 Claim 1] to derive contradiction in different cases: ε i → +∞; ε i → ε ∞ ∈ (0, +∞); ε i → 0, we obtain Claim 3.1. [3] which is based on [10].…”

“…For instances, Tosatti-Weinkove-Yang proved in [25] the C 0 loc (f −1 (Σ\S))-convergence of the metric to the generalized Kähler-Einstein metric on the base manifold Σ. In [3], the second named author and Fong considered the case when the generic fibres are biholomorphic to each other and developed a sharp parabolic Schauder estimate on cylinder using the idea of Hein-Tosatti in [10], and thus confirmed the above conjecture in the locally product case. More recently, Jian and Song [13] considered the case when m+n = 3 and proved that the Ricci curvature is uniformly locally bounded.…”

“…More recently, Jian and Song [13] considered the case when m+n = 3 and proved that the Ricci curvature is uniformly locally bounded. For further discussions, we refer interested readers to [1,3,4,6,7,9,13,20,22,25] and the references therein.…”

Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

“…known to be the case when the generic fiber is a (finite quotient of a) torus. Recently, in [15], the higher-order estimates in [29] were used to obtain the uniform bound on Ric(ω t ) when the smooth fibers are biholomorphic. Assuming this uniform bound on the Ricci curvature of the metrics along the flow, one can formulate the Conjecture for the Kähler-Ricci flow.…”