In this paper, we present a formula for pseudohermitian curvatures on bounded strictly pseudoconvex domains in with respect to the coefficients of adapted frames given by Graham and Lee in  and their structure equations. As an application, we will show that the pseudohermitian curvatures on strictly plurisubharmonic exhaustions of Thullen domains diverges when the points converge to a weakly pseudoconvex boundary point of the domain.
Let (M,𝜃) be a pseudohermitian manifold of real dimension 2n+1, i.e. M is an integrable, nondegenerate, real hypersurface in with CR structure so that 𝜃 is a 1-form satisfying for any and d𝜃 can be expressed by
where is a positive definite matrix with respect to a local basis for HM. The equivalence problem for such pseudohermitian structures was studied first by Chern using Cartan's method and later it turned out that it was related to the pseudoconformal invariants analyzed by Chern-Moser  and Tanaka . In , Webster showed that there exists a natural connection in the bundle adapted to a pseudohermitian structure, where denotes the eigenspace of the endomorphism satisfying , which defines the CR structure of M, with an eigenvalue . Moreover he gave an expression of pseudoconformal curvature tensor in terms of pseudohermitian curvature tensor.
In this paper, we present an expression of pseudohermitian curvatures on bounded strictly pseudoconvex domains in with respect to the coefficients of adapted frames given by Graham-Lee in  and their structure equations. As an application, we will show that the pseudohermitian curvatures on strictly plurisubharmonic exhaustions of Thullen domains diverge when the point converges to the weakly pseudoconvex boundary point of the domain. Our main result is
Main Theorem.For a Thullen domain
with , the pseudohermitian curvatures of the exhaustions
of with diverge as the point tends to weakly pseudoconvex boundary points along the complex line in .
2. Pseudohermitian Curvatures on Strictly Pseudoconvex Domains
In this section, we calculate the pseudohermitian curvature explicitly in terms of the coefficient of an adapted coframe for a strictly pseudoconvex bounded domain where ϕ is a defining function of Ω. Let be the adapted coframe and its dual frame given by Graham-Lee () satisfying the following.
Here R denotes the pseudohermitian curvature and denotes the sublaplacian of the pseudohermitian structure. In coordinates of , let
where and for each .
Proposition 2.1. Let be a strictly pseudoconvex domain with a defining function ϕ. Then the pseudohermitian curvature is given by