A uniqueness theorem for the singly periodic genus-one helicoid
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- by Antonio Alarcón, Leonor Ferrer and Francisco Martín PDF
- Trans. Amer. Math. Soc. 359 (2007), 2819-2829 Request permission
Abstract:
The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei’s genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic genus-one helicoid provided the existence of one symmetry.References
- H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
- Leonor Ferrer and Francisco Martín, Minimal surfaces with helicoidal ends, Math. Z. 250 (2005), no. 4, 807–839. MR 2180376, DOI 10.1007/s00209-005-0777-x
- David Hoffman, Fu Sheng Wei, and Hermann Karcher, Adding handles to the helicoid, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 77–84. MR 1193537, DOI 10.1090/S0273-0979-1993-00401-4
- David Hoffman, Hermann Karcher, and Fusheng Wei, The singly periodic genus-one helicoid, Comment. Math. Helv. 74 (1999), no. 2, 248–279. MR 1691949, DOI 10.1007/s000140050088
- David Hoffman and John McCuan, Embedded minimal ends asymptotic to the helicoid, Comm. Anal. Geom. 11 (2003), no. 4, 721–735. MR 2015173, DOI 10.4310/CAG.2003.v11.n4.a4
- D. Hoffman, M. Weber, M. Wolf. An embedded genus-one helicoid, to appear in Annals of Math.
- Joaquín Pérez, Riemann bilinear relations on minimal surfaces, Math. Ann. 310 (1998), no. 2, 307–332. MR 1602016, DOI 10.1007/s002080050150
- William H. Meeks III and Harold Rosenberg, The geometry of periodic minimal surfaces, Comment. Math. Helv. 68 (1993), no. 4, 538–578. MR 1241472, DOI 10.1007/BF02565835
- William H. Meeks III and Harold Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), no. 2, 727–758. MR 2153399, DOI 10.4007/annals.2005.161.727
- M. Weber, The genus one helicoid is embedded, 1999. Habilitationsschrift, Bonn.
Additional Information
- Antonio Alarcón
- Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain
- MR Author ID: 783655
- Email: alarcon@ugr.es
- Leonor Ferrer
- Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain
- Email: lferrer@ugr.es
- Francisco Martín
- Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain
- Email: fmartin@ugr.es
- Received by editor(s): May 4, 2005
- Published electronically: January 26, 2007
- Additional Notes: Research for this work was partially supported by MEC-FEDER grant number MTM2004-00160.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 2819-2829
- MSC (2000): Primary 53A10; Secondary 53C42
- DOI: https://doi.org/10.1090/S0002-9947-07-04093-7
- MathSciNet review: 2286058