Stability Theorems for H-Type Carnot Groups

Abstract

We introduce the H-type deviation of a step two Carnot group $\mathbb {G}$. This quantity, denoted $\delta (\mathbb {G})$, measures the deviation of $\mathbb {G}$ from the class of H-type groups. More precisely, $\delta (\mathbb {G})=0$ 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 several 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 $N({{\textbf{x}}},{{\textbf{t}}}) = (||{{\textbf{x}}}||_h^4 + 16 ||{{\textbf{t}}}||_v^2)^{1/4}$ the canonical Kaplan-type quasinorm in a step two group $\mathbb {G}$ with taming Riemannian metric $g = g_h \oplus g_v$, we show that $\mathbb {G}$ is H-type if and only if $||\nabla _0 N({{\textbf{x}}},{{\textbf{t}}})||_h^2 = ||{{\textbf{x}}}||_h^2/N({{\textbf{x}}},{{\textbf{t}}})^2$ in $\mathbb {G}{\setminus } \{0\}$. 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 the homogeneous dimension. Motivation for this work derives from a 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.