A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces

Abstract

Abstract. We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.

Conical Kähler–Ricci flows on Fano manifolds

“…1 δ 1 , ∞), such that (4.60) ψ γ ′ 1 ,ε 1 (t ′ 1 ) C 0 (M ) > L + 1. γ ′ 1 ,ε 1 (t) C 0 (M ) > L + 1.Combining Remark 3.10 with (4.58), there exists γ 1 ∈ (β, γ ′ 1 ), such that (4 62). supt∈[1,T 1 ] ψ γ 1 ,ε 1 (t) C 0 (M ) = L + 1.Then there existst 1 ∈ [1, T 1 ] such that (4.63) ψ γ 1 ,ε 1 (t 1 ) C 0 (M ) L + 7 8 .Forδ 2 = min( 1 2 , 1 T 1 +1 , ε 1 , γ 1 − β), there exist ε 2 ∈ (0, δ 2 ), γ ′ 2 ∈ (β, β + δ 2 ) and t ′ 2 ∈ [ 1 δ 2 , ∞), such that (4.64) ψ γ ′ 2 ,ε 2 (t ′ 2 ) C 0 (M ) > L + 1.Combining Remark 3.10 with (4.58) again, there exists γ 2 ∈ (β, γ ′ 2 ), such that (4.66)sup t∈[T 1 +1,T 2 ] ψ γ 2 ,ε 2 (t) C 0 (M ) = L + 1,and then there existst 2 ∈ [T 1 + 1, T 2 ] such that (4.67) ψ γ 2 ,ε 2 (t 2 ) C 0 (M ) L + 7 8 .After repeating above process, we get a subsequence ψ γ i ,ε i (t i ) with ε i ց 0, γ i ց β, t i ∈ [T i−1 +1, T i ] and t i ր ∞ satisfying(4.68) sup t∈[T i−1 +1,T i ]…”

“…These flows were first proposed in Jeffres-Mazzeo-Rubinstein's paper (see Section 2.5 in [26]), then Song-Wang (conjecture 5.2 in [48]) made a conjecture on the relation between the convergence of these flows and the greatest Ricci lower bounds of the manifolds. Then the existence, regularity and limit behavior of the conical Kähler-Ricci flows have been studied by Chen-Wang [9,10], Edwards [19,20], Liu-Zhang [34], Liu-Zhang [36,37], Nomura [39], Shen [45,46], Wang [57] and Zhang [62,63] etc.…”

Metric contraction of the cone divisor by the conical Kähler–Ricci Flow

“…While the conical Kähler-Ricci flow has been studied for many different geometric situations [11,15,24,27,34,49], little is known about the geometry of finite time singularities of the flow in general. It is reasonable to expect that singularities of the conical Kähler-Ricci flow may behave similarly to those of the Kähler-Ricci flow, however the analysis of singularities along the conical Kähler-Ricci flow is complicated by the presence of the cone divisor.…”

Cusp Kähler–Ricci flow on compact Kähler manifolds

“…Song-Wang made some conjectures on the relation between the convergence of conical Kähler-Ricci flow and the greatest Ricci lower bound of M (conjecture 5.2 in [30]). The long-time existence, regularity and limit behaviour of conical Kähler-Ricci flow have been widely studied, see the works of Liu-Zhang [19,20], Chen-Wang [4,5], Wang [38], Shen [27,28], Edwards [6], Nomura [26], Liu-Zhang [18] and Zhang [39].…”