A Scalar Curvature Bound Along the Conical Kähler–Ricci Flow

Abstract

Abstract. Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.

On a twisted conical Kähler–Ricci flow

“…Proof. This result has been obtained in [6] by using a smooth approximation. We now provide a direct argument without passing to a smooth approximation.…”

“…In fact, we will show in Proposition 2.4 that the convergence takes place exponentially fast at the level of Kähler potentials. • In Section 3, by using direct arguments on the conical equations, we prove in Proposition 3.4 that the solution is uniformly equivalent to a family of collapsing conical Kähler metric, which will immediately imply item (1) [6], our arguments don't rely on some nontrivial properties of a smooth approximation for the initial model Kähler metric (see Remark 4.2 for more comments). We point out in subsection 4.1 that arguments for Lemma 4.1 can be applied to prove Theorem 1.1.…”

“…The key Lemma 3.1 seems can not be proved by the usual smooth approximation arguments in e.g. [6,17]. A key point is that, while χ * has an upper bound for bisectional curvature, it seems unclear how to approximate χ * (in a suitable sense) by a family of smooth Kähler metrics with bisectional curvature uniformly bounded from above.…”

“…Though bounding scalar curvature along the conical Kähler-Ricci flow has been studied in some different settings, see e.g. [6,21], we still would like to provide an argument for the above Lemma 4.1. Let's list in the following Remark 4.2 the main reasons for doing this.…”

“…To get an upper bound for F ǫ , one may like to apply the maximum principle in (4.26) (see last step in [6,Section 7] or [21,Section 6]). Note that, arguing in such way, since the maximal value may be achieved at t = 0, one may need a uniform bound for |∇u ǫ |(0) = |∇ log…”