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On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions

Xiaojun Huang (1994)

Annales de l'institut Fourier

In this paper, we show that if M 1 and M 2 are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if f is a holomorphic mapping defined near a neighborhood of M 1 so that f ( M 1 ) ⊂ M 2 , then f is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for mappings...

On the maximum modulus theorem for nonanalytic functions in several complex variables.

Mario O. González Rodríguez (1981)

Revista Matemática Hispanoamericana

Let w = f(z1, ..., zn) = u(x1, ..., yn) + iv(x1, ..., yn) be a complex function of the n complex variables z1, ..., zn, defined in some open set A ⊂ Cn. The purpose of this note is to prove a maximum modulus theorem for a class of these functions, assuming neither the continuity of the first partial derivatives of u and v with respect to xk and yk, nor the conditions fzk = 0 in A for k = 1, 2, ..., n (the Cauchy-Riemann equations in complex form).

On the multivariate transfinite diameter

Thomas Bloom, Jean-Paul Calvi (1999)

Annales Polonici Mathematici

We prove several new results on the multivariate transfinite diameter and its connection with pluripotential theory: a formula for the transfinite diameter of a general product set, a comparison theorem and a new expression involving Robin's functions. We also study the transfinite diameter of the pre-image under certain proper polynomial mappings.

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