Theory of Holomorphic Functions

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

ABSTRACT: The investigator wants to study two topics in Several Complex Variables. The first is on holomorphic embeddings. By constructing holomorphic functions,one seeks to embed complex manifolds as closed submanifolds of some complex Euclidean space of as low dimension as possible. The second is on envelopes of holomorphy. In constructions of holomorphic functions, one is faced with the restriction that sometimes they all extend past certain boundary points, for example, in higher dimensions no holomorphic function can have an isolated singularity. Isolated boundary points belong to the envelope of holomorphy.The investigator will study, more generally, envelopes of holomorphy. In particular, there is a great need for investigation of a large variety of examples. Complex analysis is a basic tool in many fields of mathematics and in other sciences. Even problems described with real numbers are often dealt with more naturally after extending into the complex domain. A foundational problem of the subject is how to construct holomorphic functions with required properties. This basic problem has motivated much of the work of the proposer in the past and is also the motivation for this proposal. For example how can one find a holomorphic function tending to infinity at a certain boundary point, or has some other singularity there?