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Intrinsic structure of minimal discs in metric spaces. (English) Zbl 1378.49047

Summary: 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.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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