标题： Kähler锥度量空间中的测地线 显示英文标题

标题： Geodesics in the space of Kähler cone metrics

摘要： 在本文中，我们研究了凯勒锥度量空间中的测地线方程的狄利克雷问题$\mathcal H_\b$；这等价于一个齐次复蒙日-安培方程，其边界值由具有锥奇点的凯勒度量组成。我们的方法涉及将唐纳森\cite{MR2975584}中定义的空间推广到带有边界的凯勒流形的情况；此外，我们引入了一个子空间$\mathcal H_C$of$\mathcal H_\b$，我们通过规定适当的几何条件来定义它。我们的主要结果是边界值位于$\mathcal H_C$中的$C^{1,1}_\b$测地线的存在性、唯一性和正则性。 此外，我们证明这样的测地线是 under the $C^{1,1}_\b$-范数下的一系列$C^{2,\a}_\b$近似测地线的极限。 作为几何应用，我们证明了$\mathcal H_C$的度量空间结构。 显示英文摘要

摘要： 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 $\mathcal H_C$.