The Conical Kähler–Ricci Flow with Weak Initial Data on Fano Manifolds

Abstract

In this paper, we prove the long-time existence and uniqueness of the conical Kähler-Ricci flow with weak initial data which admits L p density for some p > 1 on Fano manifold. Furthermore, we study the convergence behavior of this flow.

Conical Kähler–Ricci flows on Fano manifolds

“…When µ γ is negative or zero, Chen-Wang [10] proved that the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with Ricci curvature µ γ and cone angle 2πγ along D. When µ γ > 0 is sufficiently small and λ 1, Li-Sun (see section 2.3 in [32], when λ = 1, see also Berman's work [1] and Jeffres-Mazzeo-Rubinstein's work [26]) proved that the Log Mabuchi energy M µγ is proper by using its definition and the property that the Log α-invariant is positive. Then the convergence of the conical Kähler-Ricci flows (CKRF µγ ) follows from the arguments in [37]. In other µ γ > 0 cases, there are obstacles.…”

“…. By the arguments in section 3 of [37], when ε tend to 0, the limit flows of (T KRF β µγ ,ε ) are the twisted conical Kähler-Ricci flows…”

“…In [37], under the assumptions that there exists a conical Kähler-Einstein metric with Ricci curvature µ γ and cone angle 2πγ along D and no nontrivial holomorphic vector fields tangent to D, we deduced their convergence. If there exists a conical Kähler-Einstein metric ω ϕ β with cone angle 2πβ along D, then ω ϕ β obtains the minimum of the Log Mabuchi energy M µ β (Corollary 2.10 in [32]), and thus this energy is bounded from blow.…”

“…Then by using the arguments in [37], we get the convergence of the conical Kähler-Ricci flows (CKRF µ β ). Instead of using the properness of M µ β in [36,37], here we only use the weaker condition that M µ β is bounded from below. Theorem 1.9.…”

“…Then the convergence of the conical Kähler-Ricci flows (CKRF µγ ) with γ ∈ (β−δ, β+δ) follows from the arguments in [36,37].…”