Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension

摘要

Roughly speaking, a d-dimensional subset of R(n) is minimizing if arbitrary211u001edeformations of it (in a suitable class) cannot decrease its d-dimensional 211u001evolume. For quasiminimizing sets one allows the mass to decrease, but only in a 211u001econtrolled manner. To make this precise we follow Almgren's notion of 'restricted 211u001esets'. Graphs of Lipschitz mappings f : R(d) -> R(n-d) are always 211u001equasiminimizing, and Almgren showed that quasiminimizing sets are rectifiable. 211u001eHere we establish uniform rectifiability properties of quasiminimizing sets, 211u001ewhich provide a more quantitative sense in which these sets behave like Lipschitz 211u001egraphs.