Linearized pseudo-Einstein equations on the Heisenberg group

Abstract

We study the pseudo-Einstein equation R 1 1 ¯ = 0 on the Heisenberg group H 1 = C × R . We consider first order perturbations θ ϵ = θ 0 + ϵ θ and linearize the pseudo-Einstein equation about θ 0 (the canonical Tanaka–Webster flat contact form on H 1 thought of as a strictly pseudoconvex CR manifold). If θ = e 2 u θ 0 the linearized pseudo-Einstein equation is Δ b u − 4 | L u | 2 = 0 where Δ b is the sublaplacian of ( H 1 , θ 0 ) and L ¯ is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain Ω ⊂ H 1 by applying subelliptic theory i.e. existence and regularity results for weak subelliptic harmonic maps. We determine a solution u to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that u ( x ) → − ∞ as | x | → + ∞ .