Collapsing limits of the Kähler–Ricci flow and the continuity method

Abstract

We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kähler metric on the total space of a Fano fibration with one dimensional base converges in Gromov-Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kähler-Einstein current on a Riemann surface is compact.

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

“…The conjecture was first verified by Gross-Wilson in [24] when f : M −→ N is an elliptically-fibered K3 surface with only I 1 -singular fibers. When the base is a curve, the conjecture was verified by [21] (see [80] also). If the canonical bundle is nef and big, the Gromov-Hausdorff limit was identified in [74].…”

“…The partial second-order estimate is motivated by the study of collapsing problems in Kähler geometry (see, e.g., [1,6,14,22,23,28,29,32,33,34,35,41,42,47,59,53,54,55,56,58,63,64,66,67,68,69,71,74,80]), as well as the study of canonical metrics in Kähler geometry and the behavior of the Kähler-Ricci flow. More specifically, the estimate in Theorem A plays a crucial role in establishing the following conjectural picture for collapsed Gromov-Hausdorff limits of Ricci-flat Kähler metrics which were originally proposed in [66,67], inspired by [24,38,39] and has been intensively studied since:…”

On the Hölder estimate of Kähler-Ricci flow

“…We focus on the case when 0 < dim Σ < dim X, and we let dim C Σ = m and dim C X = m + n, so that the Calabi-Yau fibers have complex dimension n. Under semi-ample assumption, the canonical line bundle is nef and hence the flow exists on X × [0, +∞) by the works of [2,21,26]. The Kähler-Ricci flow under semi-ample canonical line bundle has been extensively studied by various authors [4,5,6,7,11,12,17,18,19,20,22,24,25,28,29,30,31]. In [17,18], Song-Tian proved that the flow converges to a generalized Kähler-Einstein metric in the sense of measure on the base manifold Σ as t → +∞.…”

The Anomaly Flow over Riemann Surfaces

“…The metrics on the base will converge smoothly to a metric ωΣ which can be associated to the vanishing spinorial pair (Σ, ϕ). Theorem 1.1 can be compared to the phenomenon of collapsing in the Kähler-Ricci flow, as pioneered by Song-Tian [46,47] and further explored by several others [53,18,48,55,22,59]. In this case, there is a general theory of collapsing of Calabi-Yau fibrations over Kähler manifolds B.…”