Smooth approximations of the conical Kähler–Ricci flows

Abstract

In this note, we show that the conical Kähler-Ricci flows introduced in [10] exist for all time t ∈ [0, ∞) in the weak sense as in Definition 1.2. As a key ingredient of the proof, we show that a conical Kähler-Ricci flow is actually the limit of a sequence of smooth Kähler-Ricci flows. has a definite sign. Suppose ω 0 is a smooth Kähler metric in

Stability of Ricci de Turck flow on singular spaces

“…In the setting of Kähler manifolds, Kähler-Ricci flow reduces to a scalar Monge Ampere equation and has been studied in case of edge singularities in connection to the recent resolution of the Calabi-Yau conjecture on Fano manifolds by Donaldson [Don12] and Tian [Tia15], see also Jeffres, Mazzeo and Rubinstein [JMR16]. Kähler-Ricci flow in case of isolated conical singularities is geometrically, though not analytically, more intricate than edge singularities and has been addressed by Chen and Wang [ChWa15], Wang [Wan16], as well as Liu and Zhang [? ].…”

On a twisted conical Kähler–Ricci flow

“…For any T < ∞, we know ϕ is uniformly bounded on [0, T ] (see [24,40]). Therefore, the maximal value of ϕ λ on X × [0, T ] must achieved at some (x T , t T ) with x T ∈ X \ D. If t T = 0, then ϕ λ (x T , t T ) is bounded from above by a constant independent of T ; otherwise t T > 0, we can apply maximum principle to (2.11) at (x T , t T ) to see that…”

“…Proof of Lemma 4.1. We need to use a smooth approximation for the conical equations (1.8) and (2.3), introduced in [17,40] and also used in e.g. [24,6].…”

“…). Also note that ϕ ǫ and ∂ t ϕ ǫ are uniformly bounded on X × [0, t 0 ] (see [24,40]). In the above arguments for Lemma 4.1, we replace Y by X, f by the identity map X → X and χ byω t 0 , and then those arguments apply and Theorem 1.1 follows.…”

The conical complex Monge–Ampère equations on Kähler manifolds

“…Guan [19,20,21], Liu-Zhang [31], Phong and Sturm et al [34,35,36,37], G. Sźekelyhidi [38], Sźekelyhidi-Tosatti [39], G. Tian [43,45], Tian-Zhu [46], Y. Wang [47], Y.Q. Wang [48] and X. Zhang [50]…”