RIEMANN DOMAINS OVER BANACH-GRASSMANN MANIFOLDS

Let E be a complex Banach space with a Schauder basis and let G(E; r) be the Grassmann manifold of all r-dimensional complex linear subspaces in E. Let (ω, φ) be a Riemann domain over G(E; r) with ω ≠ G(E; r). Then we show that ω is a domain of existence if and only if ω is pseudoconvex.

On Ramified Riemann Domains

Let ϕ be a holomorphic mapping of an n-dimensional analytic space E into Cn. If ϕ is non-degenerate at every point of E, we call the pair (E, ϕ) a Riemann domain. The notion of a Riemann domain is a generalization of the notion of a concrete Riemann surface. A Riemann domain (E, ϕ) is said to be unramified if ϕ is a local homeomorphism, and to be ramified if otherwise.