Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results

Abstract

We consider for smooth pseudoconvex bounded domains Ω ⊂ ℂn of finite type as local analytic invariants on the boundary the growth orders of the Bergman kernel and the Bergman metric and the best possible order of subellipticity e1 > 0 for the\(\bar \partial - Neumann\) problem. Furthermore, we consider as local geometric invariants on ∂Ω the order of extendability, the exponent of extendability, the 1-type, and the multitype. Various new inequalities between these invariants are proved, giving in particular analytic information from geometric input. On the other hand, a careful consideration of several series of examples of such domains Ω shows that starting fromn ≥ 3 (essentially) each of these invariants is independent of the remaining ones.