Geodesics in the space of Kähler cone metrics

Abstract

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\b$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson \cite{MR2975584} to the case of Kähler 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 ${\mathcal{H}}_C$.

Improved presentation

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• Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor
• Expansion formula for complex Monge-Ampère equation along cone singularities
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• Geodesics in the space of Kahler cone metrics, II. Uniqueness of constant scalar curvature Kahler cone metrics
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• Generalized Matsushima's theorem and Kähler-Einstein cone metrics
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