On the Hölder estimate of Kähler-Ricci flow

Abstract

In this work, we study the Hölder regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we obtain a uniform spatial C α estimate along the Kähler-Ricci flow as t → +∞.

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

“…7 6 The assumption of maximal variation implies that ωWP is non-degenerate, and therefore an honest Kähler metric on Y • . 7 The point here is that since the divisor E has simple normal crossings, the local model for Y \E near E is specified by (∆ * ) k × ∆ n−k . Cadorel then explicitly verifies that one may apply the one-dimensional…”

On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

“…where ω can := ω N + i∂ ∂ϕ is a Kähler metric on N • and ϕ ∈ C 0 (N ). After earlier work in [51,22,60,35,21], it was recently shown in [13] that ω(t) → f * ω can in C α loc (M • ) as t → ∞, for any 0 < α < 1. Furthermore, in [38] it is shown that diam(M, ω(t)) C, for all t 0, and [53] shows that the metric completion (Z, d Z ) of (N • , ω can ) is a compact metric space, which is homeomorphic to N when this is smooth.…”