Seminar Talk: "Plateau's Problem and Cohomological Spanning"

Harrison Pugh (Stony Brook University)

Abstract:
Plateau's problem is to find a surface with minimal area spanning a given boundary. I will discuss a new theory in which the usual homological definition of span is replaced with a cohomological one. If M is a connected, oriented compact manifold of dimension n-2 in R^n, we say a compact set X "spans" M if X intersects every Jordan curve whose linking number with M is one. This definition generalizes to more general boundaries, and to higher codimension. Let S be the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of "size," we prove:
There exists an X_0 in S with smallest size. Any such X_0 contains a "core" Y_0 with the following properties: It is a subset of the convex hull of M and is a.e. a real analytic (n-1)-dimensional minimal submanifold. Furthermore, Y_0 supports a de Rham current S_0 whose boundary has support M. If n=3, then Y_0 has the local structure of a soap film.
A key new idea in this approach is that of a "film chain." There are interesting parallels between physical properties of actual soap films and mathematical properties of their film chain models. This work is joint with J.Harrison.

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