Volume 21, issue 5 (2017)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
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

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.

Calabi flow, Kähler metrics, complex Monge–Ampère equations
Mathematical Subject Classification 2010
Primary: 53C55
Secondary: 32W20, 32U05
Received: 19 November 2015
Revised: 10 July 2016
Accepted: 22 October 2016
Published: 15 August 2017
Proposed: Gang Tian
Seconded: John Lott, Bruce Kleiner
Robert J Berman
Chalmers University of Technology and University of Gothenburg
SE-412 96 Gothenburg
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
Département de Mathématiques
Université Paris-Sud
Bâtiment 425
Bureau 144
91405 Paris Orsay Cedex