Quasiminimal surfaces of codimension 1 and John domains

Abstract

We study codimension 1 quasiminimizing surfaces in R n , and establish uniform rectifiability and other geometric properties of these surfaces. For instance, their complementary components must be John domains. In fact we give a complete characterization of quasiminimizers. As an application we show that sets which are not too large and which separate points in a definite way must have a large uniformly rectifiable piece. In this way we use area quasiminimizers to solve a problem in geometric measure theory.

The quantitative isoperimetric inequality and related topics

“…Therefore it is not surprising that although quasiminimality is a very weak property, yet area quasi-minimizers have some kind of mild regularity. This is indeed the content of the next result which was first proved by David and Semmes in [51] and then extended by Kinnunen et al [88] to the metric spaces setting.…”

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

“…Also see [DS2] for a use of this flexibility in codimension 1 and in a slightly different context. Finally observe that Theorem 3.10 (the uniform rectifiability of E) is a little less far from optimality in the context of quasiminimal sets, because Lipschitz graphs and bilipschitz images of d-planes are easily shown to be quasiminimal (the class of quasiminimal sets is closed under bilipschitz mappings).…”

“…For such separation problems, we can state the problem in terms of the separated components, use the compactness properties of BV to find sets of finite perimeter that minimize, and obtain minimizers for our initial problem. We do this in [DS2], for instance, but this is very classical.…”

Some topics concerning homeomorphic parameterizations

“…Concerning variational problems, one might also keep in mind the approaches of [DaviS2], [DaviS3] (and some earlier ideas of Morel and Solimini [MoreS]). For these one does not necessarily work directly with mappings or potential parameterizations of sets, and in particular one may allow sets themselves to be variables in the minimization (rather than mappings between fixed spaces).…”