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Function Theory and Holomorphic Maps on Symmetric Products of Planar Domains

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Abstract

We show that the \(\overline{\partial }\)-problem is globally regular on a domain in \({\mathbb {C}}^n\), which is the \(n\)-fold symmetric product of a smoothly bounded planar domain. Remmert–Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from Cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary.

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Acknowledgments

We would like to thank Włodzimierz Zwonek and Armen Edigarian for helpful information and Łukasz Kosiński for telling us about his result on the boundary geometry of the symmetrized bidisc (Lemma 5.4 below). Debraj Chakrabarti would like to thank Diganta Borah for his invitation to visit IISER Pune, and colleagues there for useful comments and discussions on the subject of this paper. Sushil Gorai would like to thank Stat–Math Unit, Indian Statistical Institute, Bengaluru Centre for providing a congenial atmosphere for research. We would also like to express our special gratitude to the Dean of TIFR-CAM, Bengaluru, Prof. Mythily Ramaswamy for facilitating our collaboration. Without her support (including the provision of chauffeured cars for travel in Bengaluru) this paper would never have been written. Sushil Gorai was supported by an INSPIRE Faculty Fellowship awarded by DST, Government of India.

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Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, Mt. Pleasant, MI, 48859, USA

    Debraj Chakrabarti

  2. Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bengaluru, 560059, India

    Sushil Gorai

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  1. Debraj Chakrabarti
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  2. Sushil Gorai
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Correspondence to Debraj Chakrabarti.

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Communicated by Mei-Chi Shaw.

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Chakrabarti, D., Gorai, S. Function Theory and Holomorphic Maps on Symmetric Products of Planar Domains. J Geom Anal 25, 2196–2225 (2015). https://doi.org/10.1007/s12220-014-9509-y

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  • DOI: https://doi.org/10.1007/s12220-014-9509-y

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