The asymptotic expansion of a CR invariant and Grauert tubes
type: Item
instance of: scholarly article
title: The asymptotic expansion of a CR invariant and Grauert tubes
zbMATH Open document ID: 0848.32004
DOI: 10.1007/BF01446285
author: Su-Jen Kan
published in: Mathematische Annalen
publication date: 1996-10-28T00:00:00Z
full work available at URL: https://eudml.org/doc/165395
review text: 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.
Mathematics Subject Classification ID: 32V99
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32G07
zbMATH DE Number: 840214
zbMATH Keywords: CR-invariant
zbMATH Keywords: CR-manifold
zbMATH Keywords: CR-structure
zbMATH Keywords: Monge-Ampère model
zbMATH Keywords: Grauert tube
MaRDI profile type: Publication
Wikidata QID: Q125573690
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description: scientific article; zbMATH DE number 840214