Intrinsic Lipschitz graphs and vertical β-numbers in the Heisenberg group

Abstract

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group H. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 1990s. The theory in H has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in R3. Our main object of study are the intrinsic Lipschitz graphs in H, introduced by B. Franchi, R. Serapioni, and F. Serra Cassano in 2006. We claim that these 3-dimensional sets in H, if any, deserve to be called quantitatively 3-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical β-numbers. Conversely, extending a result of David and Semmes from Rn, we prove that a 3-Ahlfors-David regular subset in H, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.