Envelopes of holomorphy and holomorphic discs

TL;DR

This paper characterizes the envelope of holomorphy in two-dimensional Stein manifolds using equivalence classes of analytic discs, revealing its structure and implications for analytic continuation and fundamental group representations.

Contribution

It provides a new description of the envelope of holomorphy via analytic discs and explores its consequences for analytic continuation and fundamental group elements.

Findings

Envelope of holomorphy corresponds to connected components of analytic disc classes.

Analytic continuation can be achieved with a single application of the continuity principle.

Fundamental group elements can be represented by boundaries of analytic discs in certain contexts.

Abstract

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the continuity principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.