We derive conditions under which a holomorphic mapping of a taut Riemann surface must be an automorphism. This is an analogue involving invariant distances of a result of H. Cartan. Using similar methods we prove an existence result for 1-dimensional holomorphic retracts in a taut complex manifold.

Let $𝕋$ be a complex one-dimensional torus. We prove that all subsets of $𝕋$ with finitely many boundary components (none of them being points) embed properly into ${ℂ}^2$. We also show that the algebras of analytic functions on certain countably connected subsets of closed Riemann surfaces are doubly generated.

We investigate the geometry of universal embedding spaces for compact almost-complex manifolds of a given dimension, and related constructions that allow for an extrinsic study of the integrability of almost-complex structures. These embedding spaces were introduced by J-P. Demailly and H. Gaussier, and are complex algebraic analogues of twistor spaces. Their goal was to study a conjecture made by F. Bogomolov asserting the “transverse embeddability” of arbitrary compact complex manifolds into foliated...

First, we give some characterizations of J-hyperbolic points for almost complex manifolds. We apply these characterizations to show that the hyperbolic embeddedness of an almost complex submanifold follows from relative compactness of certain spaces of continuous extensions of pseudoholomorphic curves defined on the punctured unit disc. Next, we define uniformly normal families of pseudoholomorphic curves. We prove extension-convergence theorems for these families similar to those obtained by Kobayashi,...

We characterize intrinsically two classes of manifolds that can be properly embedded into spaces of the form ${\mathbb{P}}^N\setminus {\mathbb{P}}^{N-q}$. The first theorem is a compactification theorem for pseudoconcave manifolds that can be realized as $\overline{X}\setminus (\overline{X}\cap {\mathbb{P}}^{N-q})$ where $\overline{X}\subset {\mathbb{P}}^N$ is a projective variety. The second theorem is an embedding theorem for holomorphically convex manifolds into ${\mathbb{P}}^1\times {ℂ}^N$.