Universality of the Future Chronological Boundary

The purpose of this note is to establish, in a categorical manner, the universality of the Geroch-Kronheimer-Penrose causal boundary when considering the types of causal structures that may profitably be put on any sort of boundary for a spacetime. Actually, this can only be done for the future causal boundary (or the past causal boundary) separately; furthermore, only the chronology relation, not the causality relation, is considered, and the GKP topology is eschewed. The final result is that there is a unique map, with the proper causal properties, from the future causal boundary of a spacetime onto any ``reasonable" boundary which supports some sort of chronological structure and which purports to consist of a future completion of the spacetime. Furthermore, the future causal boundary construction is categorically unique in this regard.Comment: 25 pages, AMS-TeX; 2 figures, PostScript (separate); captions (separate); submitted to Class. Quantum Grav, slight revision: bottom lines legible, figures added, expanded discussion and example

Semicontinuity of the Automorphism Groups of Domains with Rough Boundary

Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in \CC^n, $n > 1$, with Lipschitz boundary, but it holds for domains in \CC^1 with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergman curvature, stability properties for the Bergman kernel, and also some ideas about equivariant embeddings

On Limits of Sequences of Holomorphic Functions

We study functions which are the pointwise limit of a sequence of holomorphic functions. In one complex variable this is a classical topic, though we offer some new points of view and new results. Some novel results for solutions of elliptic equations will be treated. In several complex variables the question seems to be new, and we explore some new avenues.Comment: one figur