The asymptotic expansion of a CR invariant and Grauert tubes

Abstract

manifolds, the abstract models of real hypersurfaces in complex manifolds, are 2n + 1 dimensional manifolds M with a codimension one subbundle H of the tangent bundle, which carries a complex structure. The CR refers to Cauchy-Riemann because for M C C n+l, the subbundle H consists of induced Cauchy-Riemann operators. There is a wealth of geometry and analysis associated with these structures, especially when the manifolds are strictly pseudoconvex. For example, two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are equivalent. A fundamental problem in geometry is to find computable invariants associated with the structures. The global invariant we will consider in this paper is the Chem-Simons type invariant/~ discovered by Bums and Epstein [B-E 1]. It is a real-valued global invariant of a compact 3dimensional strictly pseudo-convex manifold whose holomorphic tangent bundle is trivial. (Cheng and Lee independently found this invariant, and extend the definition of B-E invariant to a relative invariant on an arbitrary compact 3-dimensional manifold, cf. [C-L].) We will evaluate this g asymptotically on the boundary of small Grauert tubes. Before posing the question in a more precise form we will first say a few workds about Grauert tubes. Let X be a real analytic manifold. Then every coordinate patch U C R n can be thickened to obtain an open set r c C n. Since the coordinate changes of X are real analytic maps, by taking power series expansions and by shrinking C U to get convergence, they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition