We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B 4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.

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Let M \subset {\complex}^n be\ud a complex n-dimensional Hermitian symmetric space endowed with\ud the hyperbolic form \omega_{hyp}. Denote by (M^*, \omega_{FS}) the compact dual of (M, \omega_{hyp}), where\omega_{FS} is the Fubini--Study form on M^*. Our first result\ud is Theorem 1 where, with the aid of the theory of Jordan triple systems, we construct an explicit {\em symplectic\ud duality}, namely a diffeomorphism \Psi_M: M\rightarrow\ud {\real}^{2n}={\complex}^n\subset M^* satisfying\ud \Psi_M^*\omega_0=\omega_{hyp} and\ud \Psi_M^*\omega_{FS}=\omega_0 for the pull-back of \Psi_M, where \omega_0\ud is the restriction to M of the flat Kaehler form of\ud the Hermitian\ud positive Jordan triple system associated to M.\ud Amongst other properties of the\ud map \Psi_M, we also show that it takes (complete) complex and\ud totally geodesic submanifolds of $M$ through the origin to complex\ud linear subspaces of {\complex}^n. As a byproduct of the proof of\ud Theorem \ref{mainteor} we get an interesting characterization\ud of the Bergman form of a Hermitian\ud symmetric space in terms of its restriction to classical complex\ud and totally geodesic submanifolds passing through the origin

We prove that a homogeneous bounded domain admits a Berezin quantizatio

Abstract. This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that αg is not balanced for all α > 0.