Intrinsic Lipschitz Graphs and Vertical β-Numbers in the Heisenberg Group
| Autor: | Tuomas Orponen, Vasileios Chousionis, Katrin Fässler |
|---|---|
| Přispěvatelé: | Department of Mathematics and Statistics |
| Rok vydání: | 2019 |
| Předmět: |
osittaisdifferentiaaliyhtälöt
28A75 (Primary) 28C10 35R03 (Secondary) SETS General Mathematics 010102 general mathematics 16. Peace & justice Lipschitz continuity 01 natural sciences Travelling salesman problem Combinatorics Mathematics - Metric Geometry Mathematics - Classical Analysis and ODEs TRAVELING SALESMAN PROBLEM 0103 physical sciences 111 Mathematics Heisenberg group Mathematics::Metric Geometry mittateoria 010307 mathematical physics RECTIFIABILITY 0101 mathematics Mathematics |
| Zdroj: | American Journal of Mathematics. 141:1087-1147 |
| ISSN: | 1080-6377 |
| DOI: | 10.1353/ajm.2019.0028 |
| Popis: | 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. Comment: 56 pages, one figure. v3: incorporated referee suggestions, to appear in Amer. J. Math |
| Databáze: | OpenAIRE |
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