CHAPTER 11 - Flat Chains Modulo v, Varifolds, and (M, ε, δ)-Minimal Sets

Abstract

A number of alternative spaces of surfaces have been developed in geometric measure theory as required for theory and applications. This chapter gives brief descriptions of flat chains modulo v, varifolds, and (M, ɛ, δ)-minimal sets. One way to treat nonorientable surfaces and more general surfaces is to work modulo 2 or more generally modulo ν for any integer ν ≥ 2. Most of the concepts and theorems on rectifiable currents have analogs for rectifiable currents modulo v: mass, flat norm, the deformation theorem, the compactness theorem, existence theory, and the approximation theorem. Varifolds provide an alternative perspective to currents for working with rectifiable sets. Varifolds carry no orientation and, hence, there is no cancellation in the limit and no obvious definition of boundary. Perhaps the best model of soap films is provided by the (M, ɛ, δ)-minimal sets of Almgren. Three basic properties of (M, ɛ, δ)-minimal sets are rectifiability, monotonicity, and the existence of an (M, ɛ, δ)-minimal tangent cone at every point.