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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Convexity of the extended K-energy and the large time behavior of the weak Calabi flow

Robert J Berman, Tamás Darvas and Chinh H Lu

Geometry & Topology 21 (2017) 2945–2988
Abstract

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.

Keywords
Calabi flow, Kähler metrics, complex Monge–Ampère equations
Mathematical Subject Classification 2010
Primary: 53C55
Secondary: 32W20, 32U05
References
Publication
Received: 19 November 2015
Revised: 10 July 2016
Accepted: 22 October 2016
Published: 15 August 2017
Proposed: Gang Tian
Seconded: John Lott, Bruce Kleiner
Authors
Robert J Berman
Chalmers University of Technology and University of Gothenburg
SE-412 96 Gothenburg
Sweden
Tamás Darvas
Department of Mathematics
University of Maryland
College Park, MD 20742-4015
United States
Chinh H Lu
Chalmers University of Technology and University of Gothenburg
Kaptensgatan 28D
SE-414 59 Göteborg
Sweden
Département de Mathématiques
Université Paris-Sud
Bâtiment 425
Bureau 144
91405 Paris Orsay Cedex
France