Geodesics in the space of Kaehler cone metrics, I

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.