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A global characterization of tubed surfaces in $ \mathbb{C}^2$


Author: Michael Bolt
Journal: Proc. Amer. Math. Soc. 138 (2010), 2771-2777
MSC (2010): Primary 32V40; Secondary 53C45
DOI: https://doi.org/10.1090/S0002-9939-10-10449-3
Published electronically: April 8, 2010
MathSciNet review: 2644891
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Abstract: Let $ M^3\subset \mathbb{C}^2$ be a three times differentiable real hypersurface. The Levi form of $ M$ transforms under biholomorphism, and when restricted to the complex tangent space, the skew-Hermitian part of the second fundamental form transforms under Möbius transformations. The surfaces for which these forms are constant multiples of each other were identified in previous work, provided the constant is not unimodular. Here it is proved that if the surface is assumed to be complete and if the constant is unimodular, then the surface is tubed over a strongly convex curve. The converse statement is true, too, and is easily proved.


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Additional Information

Michael Bolt
Affiliation: Department of Mathematics and Statistics, Calvin College, 1740 Knollcrest Circle SE, Grand Rapids, Michigan 49546-4403
Email: mbolt@calvin.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10449-3
Received by editor(s): September 18, 2009
Published electronically: April 8, 2010
Additional Notes: This is based on work supported by the National Science Foundation under Grant No. DMS-0702939 and by Calvin College through a Calvin Research Fellowship.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2010 American Mathematical Society

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