We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple$f$of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with$f$when$f$defines a complete intersection.

We extend a recent result of Avelin, Hed, and Persson about approximation of functions f that are plurisubharmonic on a domain Ω and continuous on ˙Ω, with functions that are plurisubharmonic on (shrinking) neighborhoods of ˙Ω. We show that such approximation is possible if the boundary of Ω is C0 outside a countable exceptional set E⊂∂Ω. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous u, approximation is possible under less restrictive conditions on E. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

Let X1 and X2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions 2m − 1 and 2n − 1 in Cm+1 and Cn+1, respectively. We introduce the Thom- Sebastiani sum X = X1⊕X2 which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in Cm+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1 for all n > 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X1 ⊕X2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if X = X1 ⊕ X2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X1 and X2 provided that X2 admits a transversal holomorphic S1-action.

Let $T(S)$ be the Teichmuller space over a hyperbolic Riemann surface $S$. A geodesic disk in $T(S)$ is defined the image of an isometric embedding of the Poincar\'e disk into $T(S)$. In this paper, it is shown that for any non-Strebel point $\tau\in T(S)\backslash\{[0]\}$, there are infinitely many geodesic disks containing the straight line $\{[t\mu]:\,t\in (-1/k,1/k)\}$ where $\mu$ is an extremal representative of $\tau$ with $\|\mu\|_\infty=k$. An infinitesimal version is also obtained.

In this paper, we develop an approach to the study of compact K~ hler manifolds which admit mappings of everywhere maximal rank into quotients of polydiscs, eg into Riemann surfaces or products of them. One main tool will be a detailed study of the harmonic maps in the corresponding homotopy classes (for definition and general properties of harmonic maps between Riemannian manifolds see [3]). Starting with a result of Siu, we prove in Sect. 2 that the local level sets of the components of these mappings are analytic subvarieties of the domain. This, together with a generalization of the similarity principle of Bers and Vekua which is proved in the appendix and a residue argument, enables us to give conditions involving the Chern and K~ ihler classes of the considered manifolds, under which this harmonic map is of maximal rank everywhere and, in case domain and image have the same dimension, in