49,586 research outputs found
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
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, , 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
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