Geodesics in the space of K\"ahler cone metrics

In this paper, we study the Dirichlet problem of the geodesic equation in the space of K\"ahler cone metrics $\mathcal H_\b$; 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 space defined in Donaldson \cite{MR2975584} to the case of K\"ahler manifolds with boundary; moreover we introduce a subspace $\mathcal H_C$ of $\mathcal H_\b$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_\b$ geodesics whose boundary values lie in $\mathcal H_C$. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2,\a}_\b$ approximate geodesics under the $C^{1,1}_\b$-norm. As a geometric application, we prove the metric space structure of $\mathcal H_C$.