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- Title
Existence and soap film regularity of solutions to Plateau's problem.
- Authors
Harrison, Jenny; Pugh, Harrison
- Abstract
Plateau's problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension and the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of 'size,' we prove: There exists an in with smallest size. Any such contains a 'core' with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of -dimensional Hausdorff measure) a real analytic -dimensional minimal submanifold. If , then has the local structure of a soap film. Furthermore, set theoretic solutions are elevated to current solutions in a space with a rich continuous operator algebra.
- Subjects
PLATEAU'S problem; MINIMAL surfaces; HOMOLOGICAL algebra; SPANNING trees; HAUSDORFF measures
- Publication
Advances in Calculus of Variations, 2016, Vol 9, Issue 4, p357
- ISSN
1864-8258
- Publication type
Article
- DOI
10.1515/acv-2015-0023