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Geodesics in the space of Kähler cone metrics, I

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Calamai, Simone and Zheng, Kai (2015) Geodesics in the space of Kähler cone metrics, I. American Journal of Mathematics , 137 (5). pp. 1149-1208.

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Abstract

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics Hβ ; that is equivalent to a homogeneous complex Monge–Ampère equation whose boundary values consist of Kähler metrics with cone sin- gularities. Our approach concerns the generalization of the function space defined in Donaldson [25] to the case of Kähler manifolds with boundary; moreover we introduce a subspace HC of Hβ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of Cβ1’1 geodesics whose boundary values lie in HC. Moreover, we prove that such geodesic is the limit of a sequence of Cβ2’α approximate geodesics under the Cβ1’1 -norm. As a geometric application, we prove the metric space structure of HC.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Geodesics (Mathematics), Kählerian manifolds, Homogeneous complex manifolds
Journal or Publication Title: American Journal of Mathematics
Publisher: The Johns Hopkins University Press
ISSN: 0002-9327
Official Date: October 2015
Dates:
DateEvent
October 2015Published
21 July 2015Accepted
Volume: 137
Number: 5
Page Range: pp. 1149-1208
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”European Commissionhttp://dx.doi.org/10.13039/501100000780
UNSPECIFIEDStony Brook Universityhttp://dx.doi.org/10.13039/100007259
GR14 grant “Geometry of non-Kahler manifolds”Scuola Normale Superiorehttp://dx.doi.org/10.13039/100009093
UNSPECIFIEDGruppo Nazionale per le Strutture Algebriche, Geometriche e le loro ApplicazioniUNSPECIFIED
UNSPECIFIEDIstituto Nazionale di Alta Matematica "Francesco Severi"UNSPECIFIED

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