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 Xj (|
Xφ|
-1 Xjφ)ξ provided that
N is tangent to the characteristic direction
T of (
H1, θ
0), 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 →
Hn of a Riemannian manifold into the Heisenberg group we show that ΔΨ = 2
JT⊥ hence start a Weierstrass representation theory for minimal surfaces in
Hn.
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