Degeneration of Kähler–Ricci Solitons

Abstract

Abstract. Let (Y, d) be a GromovCHausdorff limit of n-dimensional closed shrinking Kähler-Ricci solitons with uniformly bounded volumes and Futaki invariants. We prove that off a closed subset of codimension at least 4, Y is a smooth manifold satisfying a shrinking Kähler-Ricci soliton equation. A similar convergence result for Kähler-Ricci flow of positive first Chern class is also obtained.

Kähler–Einstein metrics along the smooth continuity method

“…The arguments in [42,Proposition 11,12,13] An additional important fact used several times is that by Cheeger-ColdingTian [15], no tangent cone of the form C γ ×C n−1 can form in the Gromov-Hausdorff limit of a sequence of Kähler metrics with bounded Ricci curvature. The analogous result with the bound on Ricci curvature replaced by a bound on Ric(ω) − L v ω was shown by Tian-Zhang [47], and it also follows from the more recent work of Cheeger-Naber [16] in the general Riemannian case. With these observations the proof of the partial C 0 -estimate for solutions of (76) follows the argument in [42] closely.…”

“…In this section we briefly outline the changes that have to be made to the arguments in [42], using also techniques in Zhang [54], Tian-Zhang [47] and Phong-SongSturm [36], to prove the partial C 0 -estimate for the family of metrics ω t ∈ c 1 (M ) solving…”

“…One approach is to study the conformally related metrics g jk = e − 1 n−1 θv g jk , where g jk is the metric with Kähler form ω. This approach, similar to that used in Zhang [54] and Tian-Zhang [47] (who used the Ricci potential instead of θ v ), effectively reduces the problem to studying non-collapsed metrics with a lower Ricci curvature bound so that the theory of Cheeger-Colding can be applied. Indeed, in real coordinates the Ricci tensor of g satisfies…”

A compactness result for Fano manifolds and Kähler Ricci flows

“…The result under condition 1 is an improvement to Theorem 4.2 in [35] where the same conclusion is reached under the extra assumption that |∇∇u| is L ∞ uniformly in time. (b) The result under condition 2 is similar in spirit to the new regularity result for Ricci flow in [15] where the authors proved boundedness of the curvature tensor in a space-time cube under the condition that a heat kernel weighted entropy is close to zero.…”

Regularity of the Kähler–Ricci flow

“…An extension of the partial C 0 estimate to shrinking Kähler-Ricci solitons was given in [15]. These works are based on the compactness of CheegerColding-Tian [5] and its generalizations to solitons by [22]. We shall generalize these to the Kähler-Ricci flow on Fano manifolds in [23] under the regularity assumption of the limit M ∞ .…”