Characterization for a class of pseudoconvex domains whose boundaries having positive constant pseudo scalar curvature

Let (M, θ) be a strictly pseudoconvex pseudo-Hermitian compact hypersurface in C in the sense of Webster [34] with a pseudo-Hermitian real oneform θ on M . Let Rθ be the Webster pseudo scalar curvature for M with respect to θ. By the solution of the CR Yamabe problem given by Jerison and Lee [18], Gamara and Yacoub [10] and Gamara [9] (for n = 1), there is a pseudo-Hermitian real one-form θ so that (M, θ) has constant Webster pseudo scalar curvature Rθ. Let ρ be a defining function forM . Then θ = 1 2i (∂ρ−∂ρ) is a pseudo-Hermitian one-form for M , and any Hermitian one-form can be constructed in this way by using defining function of M . When M = S, the unit sphere in C, if ρ(z) = |z| − 1, then Rθ = n(n + 1) on M . The main purpose of the paper is to give some characterizations on ρ so that the pseudo scalar curvature Rθ is a positive constant on M if and only if M is CR-equivalent to the sphere S. Let D be a smoothly bounded pseudoconvex domain in C. Let u be a strictly plurisubharmonic exhaustion function for D (u = +∞ on ∂D). Let ρ(z) = −e−u(z). Then the Fefferman’s functional J(ρ) of ρ is defined as: