Proper superminimal surfaces of given conformal types in the hyperbolic four-space

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly on compacts in $M$ by proper conformal superminimal immersions $M\to H^4$. In particular, $H^4$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian curves in the twistor space of $H^4$

Noncritical holomorphic functions on Stein manifolds

We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).Comment: Acta Math, to appear. Remark 1. The foliation version of Theorem 4.1 was stated incorrectly in versions 1-3 of the preprint. Remark 2. Preprint versions 1-4 contained an informal statement (without proof) regarding the multi-parametric case of Theorem II. Since we are unable to justify all steps in this generality, we are withdrawing this statemen

On complete intersections

We construct closed complex submanifolds of dimension three in C^5 which are differential complete intersections but not holomorphic complete intersections. We also prove a homotopy principle concerning the removal of intersections of holomorphic mappings from Stein manifolds to complex Euclidean spaces C^d with certain closed complex subvarities of C^d.Comment: 13 pages, to be published in Annales Institut Fourier (2001