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||h4+16||t||v2)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 ||∇0N(x,t)||h2=||x||h2/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 ∇ 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.