Let (M,𝜃) be a pseudohermitian manifold of real dimension 2n+1, i.e. M is an integrable, nondegenerate, real hypersurface in ℂn+1 with CR structure HM⊂TM so that 𝜃 is a 1-form satisfying θ(X)=0 for any X∈HM and d𝜃 can be expressed by

where (hαβ¯) is a positive definite n×n matrix with respect to a local basis θ1,…,θn 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 [1] and Tanaka [4]. In [5], Webster showed that there exists a natural connection in the bundle H1,0M adapted to a pseudohermitian structure, where H1,0M denotes the eigenspace of the endomorphism J:HM⊗ℂ→HM⊗ℂ satisfying J∘J=−I, which defines the CR structure of M, with an eigenvalue −1. 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 ℂ2 with respect to the coefficients of adapted frames given by Graham-Lee in [3] 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 m>1, the pseudohermitian curvatures of the exhaustions

of Ωm with ϕ=−K(z,z)−m/(2m+1) diverge as the point tends to weakly pseudoconvex boundary points along the complex line {z1=0} in Ωm.

In this section, we calculate the pseudohermitian curvature explicitly in terms of the coefficient of an adapted coframe for a strictly pseudoconvex bounded domain Ω:={(z1,z2)∈ℂ2:ϕ(z1,z2)<0} where ϕ is a defining function of Ω. Let {θ0,θ1} be the adapted coframe and {X0,X1} its dual frame given by Graham-Lee ([3]) satisfying the following.

Here R denotes the pseudohermitian curvature and Δb:=−rαα−rαα denotes the sublaplacian of the pseudohermitian structure. In coordinates of ℂ2, let

and

where D:=A1B2−A2B1 and Aj:=∂ϕ∂zj for each j=1,2.

Proposition 2.1. Let Ω={(z1,z2)∈ℂ2:ϕ(z1,z2)<0} be a strictly pseudoconvex domain with a defining function ϕ. Then the pseudohermitian curvature is given by

Proof. By (2.1), we have

On the other hand, since

by comparing equations (2.3) and (2.4)we have

and

Since we have

by (2.5), (2.7), (2.6), we obtain

By substituting (2.8) into (2.5) and (2.7), we have

and

By (2.1) and a straightforward calculation we obtain

On the other hand, since

by comparing equations (2.11) and (2.13), we obtain

and

By considering B2×(2.14)−B1×(2.15), we have

By substituting (2.16) into (2.14), we obtain

Similarly by considering B2×(2.15)−B1× (2.15) we have

and by substituting (2.17) into (2.15), we have

Now using the expression (2.1) and (2.2), we obtain

By comparing the coefficient of dz1∧d z¯1 in the equation (2.18), one has

As a result, we complete the proof.

For m∈ℕ, let

be the Thullen domain in ℂ2 with m≥1. The diagonal of the Bergman kernel of Ωm is given by

For the detail, see [2]. Then ϕ(z,z¯):=−K(z,z)−m/(2m+1) gives a defining function of Ωm.

For a point a∈Δ:={z∈ℂ:|z|<1}, denote by ψa an automorphism of Ωm obtained by

Note that ψ−a is the inverse of ψa.

Then we have

and W:={(0,w):|w|=1}⊂∂Ωm is the set of weakly pseudoconvex boundary points of Ωm for m>1. Note that ∂Ωm∖W is the set of strictly pseudoconvex boundary points.

From now on, we will find the pseudohermitian curvature of the exhaustion

with ϵ≪1. In particular, we will observe the behavior of the pseudohermitian curvature when (0,z2)→(0,w) with |w|=1, i.e. weakly pseudoconvex points of Ωm. At z=(0,z2), we have the following:

Hence at z=(0,z2), by straightforward calculation we have

By differentiating the equation (2.5) with ∂∂z¯1, we obtain

Since A1=0 and ∂A1∂z1=0 at z=(0,z2), we have

and hence

By differentiating the equation (3.2) with ∂∂z1, we obtain

Evaluating the above equation at z=(0,z2) gives

Since

by Proposition 2.1 and equations (3.3), (3.1) we obtain

with p=m2m+1, and

Let

and

Then

Hence at z=(0,z2), we have

and by (3.4)

As a result we obtain our main theorem.