The hull of Rudin’s Klein bottle

Anderson, John

doi:10.1090/S0002-9939-2011-10998-5

The hull of Rudin’s Klein bottle
HTML articles powered by AMS MathViewer

by John T. Anderson

Proc. Amer. Math. Soc. 140 (2012), 553-560

DOI: https://doi.org/10.1090/S0002-9939-2011-10998-5

Published electronically: June 2, 2011

PDF | Request permission

Abstract:

In 1981 Walter Rudin exhibited a totally real embedding of the Klein bottle into $\mathbb {C}^2$. We show that the polynomially convex hull of Rudin’s Klein bottle contains an open subset of $\mathbb {C}^2$. We also describe another totally real Klein bottle in $\mathbb {C}^2$ whose hull has topological dimension equal to three.

References

Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99–109. MR 779607, DOI 10.1007/BF01455798
John T. Anderson, On an example of Ahern and Rudin, Proc. Amer. Math. Soc. 116 (1992), no. 3, 695–699. MR 1129870, DOI 10.1090/S0002-9939-1992-1129870-4
Andrzej Derdzinski and Tadeusz Januszkiewicz, Totally real immersions of surfaces, Trans. Amer. Math. Soc. 362 (2010), no. 1, 53–115. MR 2550145, DOI 10.1090/S0002-9947-09-04940-X
J. Duval and D. Gayet, Riemann Surfaces and Totally Real Tori, arXiv:math.cv/0910.2139
Franc Forstnerič, Complex tangents of real surfaces in complex surfaces, Duke Math. J. 67 (1992), no. 2, 353–376. MR 1177310, DOI 10.1215/S0012-7094-92-06713-5
Franc Forstnerič, Analytic disks with boundaries in a maximal real submanifold of $\textbf {C}^2$, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 1, 1–44 (English, with French summary). MR 894560, DOI 10.5802/aif.1076
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR 2150803, DOI 10.1017/CBO9780511546617
Paul Koosis, Introduction to $H_p$ spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 1669574
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
Walter Rudin, Totally real Klein bottles in $\textbf {C}^{2}$, Proc. Amer. Math. Soc. 82 (1981), no. 4, 653–654. MR 614897, DOI 10.1090/S0002-9939-1981-0614897-1
Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474, DOI 10.1007/978-0-8176-4538-0
John Wermer, Subharmonicity and hulls, Pacific J. Math. 58 (1975), no. 1, 283–290. MR 393567, DOI 10.2140/pjm.1975.58.283