Proper holomorphic mappings of positive codimension in several complex variables

A holomorphic mapping f from a bounded domain $\Omega$ in C$\sp{\rm n}$ to a bounded domain $\Omega\sp\prime$ in C$\sp{\rm N}$ is proper if and only if (f(z$\sb\nu$)) tends to the boundary b$\Omega\sp\prime$ for each sequence (z$\sb\nu$) that tends to b$\Omega$. If the domains are balls B$\sb{\rm n}$ and B$\sb{\rm N}$, Forstneric has proved that if f is sufficiently smooth up to the sphere bB$\sb{\rm n}$, then it must be rational, and Cima and Suffridge have shown that it then extends to be holomorphic past bB$\sb{\rm n}$. We prove the more general result that if (i) $\Omega$ lies on one side of a real analytic real hypersurface M in C$\sp{\rm n}$, (ii) F maps $\Omega$ holomorphically into the ball B$\sb{\rm N}$, (iii) in some neighborhood of a point p of M, F is the quotient of a holomorphic mapping by a holomorphic function, and (iv) if for each point q of M sufficiently near p, (F(z$\sb\nu$)) tends to bB$\sb{\rm N}$ as (z$\sb\nu$) tends to q within $\Omega$, then F extends to be holomorphic past M at p. We prove this extension result also for certain other target domains, e.g., generalized ellipsoids \rm\{\sum\sb{j} \vert w\sb{j}\vert\sp{2m\sb j}< 1\} in $\rm C\sp{N}.$

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Chiappari, Stephen Anthony

Contributors dc:contributor

• Miles, Joseph B.

• Mathematics

Statement dc:rights

• Copyright 1990 Chiappari, Stephen Anthony

Language dc:language

eng

Identifier

AAI9114202
(UMI)AAI9114202

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/19419