Geodesics in the space of K\"ahler cone metrics

TL;DR

This paper establishes the existence, uniqueness, and regularity of geodesics in the space of Kähler cone metrics, extending previous work to manifolds with boundary and introducing a new subspace with specific geometric conditions.

Contribution

It generalizes the space of Kähler metrics with cone singularities to manifolds with boundary and proves the existence of regular geodesics within this framework.

Findings

Existence and uniqueness of $C^{1,1}_$ geodesics.

Geodesics are limits of $C^{2,}_$ approximate geodesics.

The space $cal H_C$ forms a metric space.

Abstract

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 $\mathcal H_C$.