Geodesics in the space of Kähler cone metrics, I

Abstract

In this paper, we study the Dirichlet problem of the geodesic equation in the space of K\"ahler cone metrics ${\cal H}_{\cal B}$; that is equivalent to a homogeneous complex Monge-Amp\`ere equation whose boundary values consist of K\"ahler metrics with cone singularities. Our approach concerns the generalization of the function space defined by Donaldson to the case of K\"ahler manifolds with boundary; moreover we introduce a subspace ${\cal H}_C$ of ${\cal H}_{\cal B}$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_{\cal B}$ geodesics whose boundary values lie in ${\cal H}_C$. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2, \alpha}_{\cal B}$ approximate geodesics under the $C^{1,1}_{\cal B}$-norm. As a geometric application, we prove the metric space structure of ${\cal H}_C$.