Geometry & Topology

Convexity of the extended K-energy and the large time behavior of the weak Calabi flow

Robert Berman, Tamás Darvas, and Chinh Lu

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Let (X,ω) be a compact connected Kähler manifold and denote by (p,dp) the metric completion of the space of Kähler potentials ω with respect to the Lp–type path length metric dp. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to p is a dp–lsc functional that is convex along finite-energy geodesics. Second, following the program of J Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space (2,d2). This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the d2–metric or it d1–converges to some minimizer of the K-energy inside 2. This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. We investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. Finally, when a cscK metric exists in ω, our results imply that the weak Calabi flow d1–converges to such a metric.

Article information

Geom. Topol., Volume 21, Number 5 (2017), 2945-2988.

Received: 19 November 2015
Revised: 10 July 2016
Accepted: 22 October 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 32W20: Complex Monge-Ampère operators 32U05: Plurisubharmonic functions and generalizations [See also 31C10]

Calabi flow Kähler metrics complex Monge–Ampère equations


Berman, Robert; Darvas, Tamás; Lu, Chinh. Convexity of the extended K-energy and the large time behavior of the weak Calabi flow. Geom. Topol. 21 (2017), no. 5, 2945--2988. doi:10.2140/gt.2017.21.2945.

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