On constructing maps to Carnot groups

Wednesday, October 2, 2019 - 14:00 to 15:00

Thackeray 427

Speaker Information
Behnam Esmayli

Abstract or Additional Information

Heisenberg group, $\mathbb{H}^1$, is the usual $\mathbb{R}^3$ equipped with a new multiplication, which turns it into a Lie group, and a new metric $d_{CC}$. On bounded subsets one has
$$ |p - q | \leq d_{CC}(p,q) \leq C \sqrt{ |p-q| } \ .$$
Pansu and Gromov observed that for $\alpha > \frac{2}{3}$ there is no embedding of an open subset of $\mathbb{R}^2$ into $\mathbb{H}^1$ that is $\alpha-$Holder continuous. I will talk about some recent results in the other direction: Existence of nontrivial maps from $\mathbb{R}^2 \to \mathbb{H}^1$ that are $(\frac{2}{3} - \epsilon)$-Holder for arbitrarily small $\epsilon$.