Stability theorems for H-type Carnot groups

Abstract

We introduce the H-type deviation $\delta (\mathbb{G})$ of a step two Carnot group $\mathbb{G}$, which measures the deviation of the group from the class of Heisenberg-type groups. We show that $\delta (\mathbb{G})=0$ if and only if $\mathbb{G}$ carries a vertical metric which endows it with the structure of an H-type group. We compute the H-type deviation for several naturally occurring families of step two groups. In addition, we provide analytic expressions which are comparable to the H-type deviation. As a consequence, we establish new analytic characterizations for the class of H-type groups. For instance, denoting by $N(g)=(||x|{|}_h^4+16||t|{|}_v^2{)}^{1/4}$, $g=\exp (x+t)$, the canonical Kaplan-type quasi-norm in a step two group $\mathbb{G}$ with taming Riemannian metric $g_h\oplus g_v$, we show that $\mathbb{G}$ is H-type if and only if $||{\nabla }_{0}N(g)|{|}_h^2=||x|{|}_h^2/N(g{)}^2$ for all $g\ne 0$. Similarly, we show that $\mathbb{G}$ is H-type if and only if $N^{2-Q}$ is $\mathcal{L}$-harmonic in $\mathbb{G}\setminus \{0\}$. Here ${\nabla }_0$ denotes the horizontal differential operator, $\mathcal{L}$ the canonical sub-Laplacian, and $Q=\dim {\mathfrak{v}}_1+2\dim {\mathfrak{v}}_2$ the homogeneous dimension of $\mathbb{G}$, where ${\mathfrak{v}}_1\oplus {\mathfrak{v}}_2$ is the stratification of the Lie algebra. It is well-known that H-type groups satisfy both of these analytic conclusions. The new content of these results lies in the converse directions. Motivation for this work comes from a longstanding conjecture regarding polarizable Carnot groups. We formulate a quantitative stability conjecture regarding the fundamental solution for the sub-Laplacian on step two Carnot groups. Its validity would imply that all step two polarizable groups admit an H-type group structure. We confirm this conjecture for a sequence of anisotropic Heisenberg groups.

32 pages

References in corpus (1)

• Sharp measure contraction property for generalized H-type Carnot groups