Asymptotics of ends of constant mean curvature surfaces with bubbletons

Kobayashi, Shimpei

doi:10.1090/S0002-9939-07-09137-X

Asymptotics of ends of constant mean curvature surfaces with bubbletons
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by Shimpei Kobayashi

Proc. Amer. Math. Soc. 136 (2008), 1433-1443

DOI: https://doi.org/10.1090/S0002-9939-07-09137-X

Published electronically: December 10, 2007

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Abstract:

A constant mean curvature surface with bubbletons is defined by the loop group action on the set of extended framings for constant mean curvature surfaces by simple factors. Classically such surfaces were obtained by the transformation of tangential line congruences, the so-called Bianchi-Bäcklund transformations. In this paper, we consider constant mean curvature surfaces with Delaunay ends in three-dimensional space forms $\mathbb R^3$, $S^3$ and $H^3$ and their surfaces with bubbletons for which the topology is preserved. We show that the ends of such surfaces are again asymptotic to Delaunay surfaces.

References

Luigi Bianchi. Vorlesungen über Differentialgeometrie. Leipzig, Berlin, Druck und Verlag von B. G. Teubner, 1910.
J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633–668. MR 1664887, DOI 10.4310/CAG.1998.v6.n4.a1
M. Kilian, S.-P. Kobayashi, W. Rossman, and N. Schmitt. Unitarization of monodromy representations and constant mean curvature trinoids in $3$-dimensional space forms J. London Math. Soc. (2), 75(3):563–581, 2007.
M. Kilian, W. Rossman, and N. Schmitt. Delaunay asymptotics of constant mean curvature surfaces in space forms via loop group methods. See http://jlms.oxfordjournals.org/cgi/ content/abstract/75/3/563
M. Kilian, N. Schmitt, and I. Sterling, Dressing CMC $n$-noids, Math. Z. 246 (2004), no. 3, 501–519. MR 2073454, DOI 10.1007/s00209-003-0587-y
Shimpei Kobayashi, Bubbletons in 3-dimensional space forms, Balkan J. Geom. Appl. 9 (2004), no. 1, 44–68. MR 2205901
Nicholas Schmitt. cmclab. http://www.gang.umass.edu/software, 2003.
I. Sterling and H. C. Wente, Existence and classification of constant mean curvature multibubbletons of finite and infinite type, Indiana Univ. Math. J. 42 (1993), no. 4, 1239–1266. MR 1266092, DOI 10.1512/iumj.1993.42.42057
Chuu-Lian Terng and Karen Uhlenbeck, Bäcklund transformations and loop group actions, Comm. Pure Appl. Math. 53 (2000), no. 1, 1–75. MR 1715533, DOI 10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.3.CO;2-L