The Umehara algebra is studied with motivation on the problem of non-existence common complex submanifolds. In this paper, we prove some new results in and obtain applications. particular, if a manifolds admits holomorphic polynomial isometric immersion to one indefinite space form, then it cannot another form different type. Other consequences include submanifolds for projective or hyperbolic ...

We present some results obtained over the years regarding the disk property for complex manifolds and its connections with pseudoconvexity. The idea to use holomorphic disks to study domains of holomorphy in C goes back all the way to F. Hartogs [12] at the beginning of the twentieth century. Hartogs’ result was extended by Osgood [19] who proved what is called ”Hartogs extension theorem” stati...

100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: holomorphic functions in a connected neighborhood V(∂Ω) of a connected boundary ∂Ω b C (n > 2) do extend holomorphically and uniquely to the domain Ω. Martinelli in the early 1940’s and Ehrenpreis in 1961 obtained a rigorous pro...

We prove an analog of the classical Hartogs extension theorem for certain (possibly unbounded) domains on coverings of Stein manifolds.

We study lineally convex domains of a special type, viz. Hartogs domains, and prove that such sets can be characterized by local conditions if they are smoothly bounded.

We exhibit a class of bounded, strongly convex Hartogs domains with realanalytic boundary which are not Lu Qi-Keng, i.e. whose Bergman kernel function has a zero.

Let ∆ ⊆ C be the open unit disc and let Σ ⊆ ∆×∆ be a compact set such that K = Σ ∪ (∂∆×∆) is a connected set. It is a classical result by Hartogs that if Σ is an analytic variety over ∆ with the boundary in ∂∆×∆, then every function holomorphic in a connected neighbourhood of K extends holomorphically to a neighbourhood of ∆ × ∆. It is proved that the same conclusion holds if Σ is a ‘continuous...

The Trichotomy Principle says that a pair of sets A and B either admits a bijection or else precisely one of these sets injects into the other. Hartogs established logical equivalence between the Trichotomy Principle and the Well-Ordering Principle. As ZF suffices to prove the Schröder-Bernstein theorem, the heart of Trichotomy lies in the existence of some injection connecting A and B (in eith...

Compactness of the Neumann operator in the ∂̄-Neumann problem is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth domain is constructed for which condition (P) fails to hold, yet the Kohn Laplacian still has compact resolvent. The main result, in contrast, is that for smoothly bounded Hartogs domains, the well-known sufficient condition (P) is equivalent to ...