The asymptotic expansion of a CR invariant and Grauert tubes
Let \(M\) be complex \(n\)-dimensional manifold and let \(X\) be a totally real submanifold, real-analytically imbedded in \(M\) with a Riemannian metric \(ds^2\). It is known that there exists a strictly plurisubharmonic nonnegative function \(\rho\) on \(M\) such that \(X = \{z \in M : \rho (z) = 0\}\). The domain \(M_\varepsilon = \rho^{-1} ([0, \varepsilon^2])\) is called the Grauert tube of radius \(\varepsilon\). The function \(\rho\) is uniquely defined if in addition the metric \(ds^2_n\) which is obtained from the Kähler form \({i \over 2} \partial \overline \partial \rho\), is compatible with \(ds^2\) and if \((\partial \overline \partial \sqrt \rho)^n = 0\). On \(M \backslash X\), Burns and Epstein defined a real-valued global CR-invariant \(\mu\) for a compact, strictly pseudoconvex 3-dimensional CR-manifold \(M\) whose holomorphic tangent bundle is trivial. This \(\mu\) can be written as \(\mu = \int_M \widetilde TC_2 (\Pi)\), where \(\widetilde TC_2 (\Pi)\) is some special form. The author proves that the natural CR structure on \(\partial M_\varepsilon\) has a trivial holomorphic tangent bundle and therefore the invariant \(\mu = \mu_\varepsilon\) is well defined on \(\partial M_\varepsilon\). Further the author proves that \(\mu_\varepsilon\) has an asymptotic expansion in the radius on the tube and that the leading and the second-order terms of this expansion are respectively the volume and the scalar curvature times some dimensional constant. At the end of the paper Riemannian manifolds of constant curvature are investigated. In particular, it is proved that the corresponding Grauert tubes enclosed by different Monge-Ampère levels cannot be beholomorphically equivalent.
scientific article; zbMATH DE number 840214