Intrinsic structure of minimal discs in metric spaces

Lytchak, Alexander¹ ; Wenger, Stefan²

¹ Mathematisches Institut, Universität Köln, Köln, Germany
² Department of Mathematics, University of Fribourg, Fribourg, Switzerland

Geometry & topology, Tome 22 (2018) no. 1, pp. 591-644.

Résumé

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces.

Détail

DOI : 10.2140/gt.2018.22.591

Classification : 49Q05, 53A10, 53C23
Keywords: minimal disc, Plateau problem

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