Regularity of minimal and almost minimal sets and cones: J. Taylor’s theorem for beginners

Abstract

Abstract. We discuss various settings for the Plateau problem, a proof of J. Taylor's regularity theorem for 2-dimensional almost minimal sets, some applications, and potential extensions of regularity results to the boundary.

A direct approach to Plateau's problem in any codimension

“…Roughly speaking they showed, in the co-dimension one case, that every time one has a class which contains "enough" competitors (namely the cone and the cup competitors, see [DGM14, Definition 1]) it is always possible to show that the infimum of the Plateau's problem is achieved by the area of a rectifiable set. They then applied this result to provide a new proof of Harrison and Pugh theorem as well as to show the existence of sliding minimizers, a new notion of minimal sets proposed by David in [Dav14,Dav13] and inspired by Almgren's (M, 0, ∞), [Alm76].…”

“…Theorem 1.5. Let H be closed in R n and C be closed by isotopy with respect to H, then: The second type of boundary condition we want to consider is the one related to the notion of "sliding minimizers" introduced by David in [Dav14,Dav13].…”

Chapter Six. Should We Solve Plateau’s Problem Again?

“…When E is 2-dimensional in R 3 , Taylor [Ta2] proved that it is locally C 1 -equivalent to a minimal cone (a plane, or a set Y or T as in Section 6.3.d), and this was partially extended to higher ambient dimensions in [Da4,5]. Also see [Da6] for a slightly more detailed survey of these results.…”

“…First, because there is still a small chance that this approach will work in some cases, by selecting carefully a nice minimizing sequence before we take any limit. See [Da6], where a short description of two recent results of this type is given, but for simpler problems where the class F(E 0 ) is not given in terms of boundary pieces j as above, but of softer topological conditions. Also, the chances of proving existence results will probably increase if we understand better the regularity results for minimizers, all the way to the boundary.…”

“…In [Re1], Reifenberg does this by carefully cutting the hair of an initial E k . Let us say two words about a slightly different way; we refer to [Da6] for a more detailed account of the strategy and some applications.…”