Stability Theorems for H-Type Carnot Groups

We introduce the H-type deviation of a step two Carnot group 𝔾 . This quantity, denoted δ (𝔾) , measures the deviation of 𝔾 from the class of H-type groups. More precisely, δ (𝔾)=0 if and only if 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(x,t) = (||x||_h^4 + 16 ||t||_v^2)^1/4 the canonical Kaplan-type quasinorm in a step two group 𝔾 with taming Riemannian metric g = g_h ⊕ g_v , we show that 𝔾 is H-type if and only if ||∇ _0 N(x,t)||_h^2 = ||x||_h^2/N(x,t)^2 in 𝔾∖{0} . Similarly, we show that 𝔾 is H-type if and only if N^2-Q is ℒ -harmonic in 𝔾∖{0} . Here ∇ _0 denotes the horizontal differential operator, 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.