A variant of Gromov's problem on Holder equivalence of Carnot groups

作者：Jung Derek^*

来源：Journal of Mathematical Analysis and Applications, 2017, 456(1): 251-273.

DOI：10.1016/j.jmaa.2017.06.085

• Sub-Riemannian geometry
• Holder mappings
• Carnot groups
• Geometric measure theory
• Jet spaces
• Gromov conjecture

It is unknown if there exists a locally alpha-Holder homeomorphism f : R-3 -> H-1 for any 1/2. < alpha <= 2/3, although the identity map R-3 -> H-1 is locally 1/2-Holder. More generally, Gromov asked: Given k and a Carnot group G, for which a does there exist a locally alpha-Holder homeomorphism f : R-k -> G? Here, we equip a Carnot group G with the Carnot Caratheodory metric. In 2014, Balogh, Hajlasz, and Wildrick considered a variant of this problem. These authors proved that if k > n, there does not exist an injective, (2-1-)-Holder mapping f : R-k -> H-n that is also locally Lipschitz as a mapping into R2n+1. For their proof, they use the fact that Hn is purely k-unrectifiable for k > n. In this paper, we will extend their result from the Heisenberg group to model filiform groups and Carnot groups of step at most three. We will now require that the Carnot group is purely k-unrectifiable. The main key to our proof will be showing that (2-1-)-Holder maps f : R-k -> G that are locally Lipschitz into Euclidean space, are weakly contact. Proving weak contactness in these two settings requires understanding the relationship between the algebraic and metric structures of the Carnot group. We will use coordinates of the first and second kind for Carnot groups.

• 出版日期2017-12-1