The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.

Analytic Functions of Several Complex Variables

Holomorphic Extension of CR Functions, Envelopes of Holomorphy, and Removable Singularities

https://hal.archives-ouvertes.fr/hal-00125098/document

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlev\'{e} problem; metrically thin singularities of CR functions; geometrically removable singularities for solutions of the induced d-barre. Selected theorems are fully proved, while surveyed results are put in the right place in the architecture.

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not necessarily normal) X of pure dimension n >=2 provided that the regular part of D - K is connected.

/pdf/the-hartogs-extension-theorem-on-n-1-complete-complex-spaces-48y5eqor4g.pdf

The Hartogs extension theorem on (n-1)-complete complex spaces

This paper is divided into three parts. Part I develops a general, new theory (inspired by modern CR geometry) of Lie symmetries of completely integrable pde systems, viewed from their associated submanifolds of solutions. Part II constructs general combinatorial formulas for the prolongations of vector fields to jet spaces. Part III explicitly characterizes the flatness of some systems of the second order. The results presented here are original and were not published elsewhere; most formulas of Parts II and III were verified by means of Maple Release 7.

/pdf/lie-symmetries-and-cr-geometry-1dglgr1v85.pdf

Lie symmetries and CR geometry

We endeavour a systematic approach for the removal of singularities for CR functions on an arbitrary embeddable CR manifold.

On removable singularities for integrable CR functions

One hundered 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 ⋐ℂn ≥ 2) do extend holomorphically and uniquely to the domain ό. Martinelli, in the early 1940’s, and Ehrenpreis in 1961 obtained a rigorous proof, using a new multidimensional integral kernel or a short\(\bar \partial \) argument, but it remained unclear how to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906), and E. E. Levi (1911) in some special, model cases. In fact, known attempts (e.g., Osgood, 1929, Brown, 1936) struggled for monodromy against multivaluations, but failed to get the general global theorem.

Egmont Porten

Papers

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

The Hartogs extension theorem on (n-1)-complete complex spaces

On removable singularities for integrable CR functions

A morse-theoretical proof of the hartogs extension theorem