Abstract:
In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature.
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Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999
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Mese, C. The structure of singular spaces of dimension 2. manuscripta math. 100, 375–389 (1999). https://doi.org/10.1007/s002290050207
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DOI: https://doi.org/10.1007/s002290050207