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

Berman, Robert; Darvas, Tamás; Lu, Chinh

doi:10.2140/gt.2017.21.2945

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

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

DOI: 10.2140/gt.2017.21.2945

Abstract

Let $(X, ω)$ be a compact connected Kähler manifold and denote by $(ℰ^{p}, d_{p})$ the metric completion of the space of Kähler potentials $ℋ_{ω}$ with respect to the $L^{p}$ –type path length metric $d_{p}$ . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $ℰ^{p}$ is a $d_{p}$ –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 $(ℰ^{2}, d_{2})$ . 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 $d_{2}$ –metric or it $d_{1}$ –converges to some minimizer of the K-energy inside $ℰ^{2}$ . 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 $ℋ_{ω}$ , our results imply that the weak Calabi flow $d_{1}$ –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

Volume 21, issue 5 (2017)