Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intrinsic Lipschitz graphs and vertical $\beta$-numbers in the
  Heisenberg group   The purpose of this paper is to introduce and study some basic concepts of
quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In
particular, we aim to demonstrate that new phenomena arise compared to the
Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in
$\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not
play a role with similar questions in $\mathbb{R}^{3}$.
  Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$,
introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim
that these $3$-dimensional sets in $\mathbb{H}$, 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 $\beta$-numbers.
Conversely, extending a result of David and Semmes from $\mathbb{R}^{n}$, we
prove that a $3$-Ahlfors-David regular subset in $\mathbb{H}$, which satisfies
the weak geometric lemma and has big vertical projections, necessarily has big
pieces of intrinsic Lipschitz graphs.

35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin