We prove that the Bergman metric on domains satisfying condition S is complete. This implies that any bounded pseudoconvex domain with Lipschitz boundary is complete with respect to the Bergman metric. We also show that bounded hyperconvex domains in the plane and convex domains in ${ℂ}^n$ are Bergman comlete.

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing...

In this paper we study infinitesimal CR automorphisms of Levi degenerate hypersurfaces. We illustrate the recent general results of [18], [17], [15], on a class of concrete examples, polynomial models in ${ℂ}^3$ of the form $\Im w=\Re \left(P\left(z\right)\overline{Q\left(z\right)}\right)$, where $P$ and $Q$ are weighted homogeneous holomorphic polynomials in $z=(z_1,z_2)$. We classify such models according to their Lie algebra of infinitesimal CR automorphisms. We also give the first example of a non monomial model which admits a nonlinear rigid automorphism.

We study the Chern-Moser operator for hypersurfaces of finite type in ${ℂ}^2$. Analysing its kernel, we derive explicit results on jet determination for the stability group, and give a description of infinitesimal CR automorphisms of such manifolds.

This paper studies local geometry of hypersurfaces of finite multitype. Catlin’s definition of multitype is applied to a general smooth hypersurface in ${ℂ}^{n+1}$. We prove biholomorphic equivalence of models in dimension three and describe all biholomorphisms between such models. A finite constructive algorithm for computing multitype is described. Analogous results for decoupled hypersurfaces are given.