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Characterizations of Boundaries of Holomorphic 1-Chains within \(\hat{\mathbb{C}}\times\hat{\mathbb{C}}\) and \(\mathbb{C}\times\hat{\mathbb{C}}\)

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Abstract

By way of a particular Cauchy-type integral, we characterize the closed rectifiable 1-currents γ, with support contained in ℂ2 and satisfying condition A 1, that bound holomorphic 1-chains within \(\hat{\mathbb{C}}\times\hat{\mathbb{C}}\) . Also, we derive characterizations for the boundaries of holomorphic 1-chains within \(\mathbb{C}\times\hat {\mathbb{C}}\) , which yield examples of characterizations within a non-compact, non-Stein space. Additionally, we illustrate a connection between some of these characterizations and the Cauchy integral characterization of boundary values of meromorphic and holomorphic functions over a domain in ℂ.

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Authors and Affiliations

  1. Dept. of Computer Science and Mathematical Sciences, Penn State Harrisburg, 777 West Harrisburg Pike, Middletown, PA, 17057, USA

    Ronald A. Walker

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  1. Ronald A. Walker
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Correspondence to Ronald A. Walker.

Additional information

Communicated by Steven Krantz.

This work is based in part on work supported by a NSF Graduate Fellowship while the author was at the University of Michigan and work while the author was a VIGRE Ross Assistant Professor at the Ohio State University.

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Walker, R.A. Characterizations of Boundaries of Holomorphic 1-Chains within \(\hat{\mathbb{C}}\times\hat{\mathbb{C}}\) and \(\mathbb{C}\times\hat{\mathbb{C}}\) . J Geom Anal 18, 1159–1170 (2008). https://doi.org/10.1007/s12220-008-9036-9

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  • DOI: https://doi.org/10.1007/s12220-008-9036-9

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