Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary

Abstract

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold M endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure θ on M, [20]) on the total space of the canonical circle bundle S¹ → C(M) $\stackrel{\pi}{\ ightarrow}$ M (a manifold with boundary ∂C(M) = π^-1 (∂M)) and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface N = { φ = 0 } ⊂ H₁ we show that the mean curvature vector of N $\hookrightarrow$ H₁ is expressed by H = - $\frac{1}{2}$ $\sum$_j=1² X_j (|Xφ|^-1 X_jφ)ξ provided that N is tangent to the characteristic direction T of (H₁, θ₀), thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g. [7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion Ψ:N → H_n of a Riemannian manifold into the Heisenberg group we show that ΔΨ = 2 JT^⊥ hence start a Weierstrass representation theory for minimal surfaces in H_n.