Assume both X and Y are Riemann surfaces which are subsets of compact Riemann surfaces X1 and Y1, respectively, and that the set X1 −X has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on X and Y are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows...

De nition 3.1. Let X be a complex manifold. A holomorphic vector bundle of rank k over E is a complex manifold E and a holomorphic map : E ! X such that makes E ! X into a complex vector bundle of rank k, and E admits holomorphic trivializations, i.e. there’s an open cover U of X trivializing E such that for each U 2 U, there’s a biholomorphic map ' : EjU ! U C commuting with projection to U th...

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets the form $K\cup M$ a manifold $X$, where $K$ is compact $\mathscr O(U)$-convex set an Stein neighbourhood $U$ $K$, $M$ embedded submanifold and $K\cap O(M)$-convex. prove Docquier-Grauert type theorem concerning biholomorphic equivalence such sets, give sufficient conditions for existence M...