In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kaehler cone metrics $mathcal H_eta$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kaehler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson cite{MR2975584} to the case of Kaehler manifolds with boundary; moreover we introduce a subspace HC of $mathcal H_eta$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_$ geodesics whose boundary values lie in HC. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2,a}_$ approximate geodesics under the $C^{1,1}_$-norm. As a geometric application, we prove the metric space structure of HC.

Geodesics in the space of Kaehler cone metrics, I / Simone, Calamai; Kai, Zheng. - In: AMERICAN JOURNAL OF MATHEMATICS. - ISSN 0002-9327. - STAMPA. - 137:(2015), pp. 1149-1208. [10.1353/ajm.2015.0036]

Geodesics in the space of Kaehler cone metrics, I

CALAMAI, SIMONE;
2015

Abstract

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kaehler cone metrics $mathcal H_eta$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kaehler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson cite{MR2975584} to the case of Kaehler manifolds with boundary; moreover we introduce a subspace HC of $mathcal H_eta$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_$ geodesics whose boundary values lie in HC. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2,a}_$ approximate geodesics under the $C^{1,1}_$-norm. As a geometric application, we prove the metric space structure of HC.
2015
137
1149
1208
Simone, Calamai; Kai, Zheng
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/863501
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