Let \((X,g)\) be a compact Riemannian manifold and \(X_{\mathbb C} \) its canonical complexification. This complexification is realized as a subset \(T^rX:=\{v \in TX\mid \| v\| <r\}\) of the tangent bundle \(TX\) of \(X\) for some \(0<r\leq \infty\), where \(\| v\| \) denotes the norm of the tangent vector \(v\) with respect to the Riemannian metric \(g\). Then \(X\) is identified with the zero section of \(TX\), and the function \(\rho (v) :=2\| v\| ^2\) is strongly plurisubharmonic. The manifold \(T^rX\) is called the Grauert tube of radius \(r\) over \(X\).
Let now \(\Omega_\varepsilon:=\{\rho < \varepsilon^2\}\), for \(0< \varepsilon < r\sqrt{2}\). Then \(M_\varepsilon= \{ \rho = \varepsilon^2\}\) is a strongly pseudoconvex CR manifold. Let \(\iota_\varepsilon : M_\varepsilon \rightarrow T^rX\) be the embedding and \(\theta^{(\varepsilon)}\) the pseudo-Hermitian structure on \(M_\varepsilon\) defined by \(\theta_\varepsilon=\iota_\varepsilon^* (-\sqrt{-1}\,\partial \rho)\).
The Szegö kernel \(S_{\theta^{(\varepsilon)}}\) has an asymptotic expansion of the form \[ S_{\theta^{(\varepsilon)}} (z, \bar z) = \, \frac{\varphi (z)}{r(z)^n} +\psi_{\theta^{(\varepsilon)}} (z) \,\log \, r(z) \] and \(r(z)\) denotes a defining function for \(\Omega_{\varepsilon}\). The main result of the article under review is a formula for the asymptotic behavior of \(\psi_0^\varepsilon(z):=\psi_{\theta^{(\varepsilon)}} (z)\) when \(\varepsilon\) tends to zero, in the case that \(n=2\).
Theorem: Assume that \(n=2\). Then \(\psi_0^\varepsilon\) has the asymptotic expansion as \(\varepsilon \searrow 0\): \[ \psi_0^\varepsilon (U) = \sum_{l=0}^L F_l (U) \varepsilon^{2l} + O(\varepsilon^{2L+1}), \] for all \(L\geq 0\), where \(O(\varepsilon^m)\) stands for a function \(f(U)\) such that \(f(U)/\varepsilon^m\) is bounded as \(\varepsilon \searrow 0\). The coefficient \(F_0\) is given by \[ F_0(U) = -\frac{1}{120\pi^2} \left( \nabla_i\nabla_j K - \frac{1}{2} (\nabla_k\nabla^k\,K) g_{ij} \right) u^iu^j , \] where \(K\) is the Gaussian curvature, \((\nabla_i\nabla_j K)u^iu^j\) is a contraction of \(\nabla^2 K \otimes U\otimes U,\) and \(-\nabla_k\nabla^k\) is the Laplacian.