The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

中文翻译：

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度量空间中有界属的面积最小化表面

高原-道格拉斯问题要求找到一个区域，该区域最小化固定或有界属的表面，该范围跨越给定的约旦曲线在欧几里得空间中的有限集合。在本文中，我们在适当的度量空间设置中解决了这个问题，该空间允许曲线具有局部二次等距不等式。此外，我们获得了解的边界和内部Hölder规则性的连续性。我们的结果概括了Jost和Tomi-Tromba的相应结果，从黎曼流形的设置到具有局部二次等距不等式的适当度量空间的设置。在Lytchak和第二作者最近解决的适当度量空间中，跨越单条Jordan曲线的盘状表面的特殊情况对应于Plateau的经典问题。