Simultaneous unitarizability of \(SL_n\mathbb{C}\)-valued maps,and costant mean curvature \(k\)-noid monodromy. (English) Zbl 1150.53021
The authors study uniterization theorems for maps into \(SL_n\mathbb{C}\). As a consequence of their results they can construct families of constant mean curvature surfaces with arbitrarily many ends into ambient 3-dimensional space forms. This is done by using the uniterization theorem in order to solve the monodromy problem arising in the construction of constant mean curvature surfaces.
The authors also show how the class of trinoids, originally constructed in a previous paper by the same authors together with M. Kilian and S. Kobayashi [J. Lond. Math. Soc., II. Ser. 75, No. 3, 563–581 (2007; Zbl 1144.53017)], may be obtained in an easier way. Another advantage of the technique presented in this paper is that it may also be used in order to construct symmetric \(n\)-noids.
The authors also show how the class of trinoids, originally constructed in a previous paper by the same authors together with M. Kilian and S. Kobayashi [J. Lond. Math. Soc., II. Ser. 75, No. 3, 563–581 (2007; Zbl 1144.53017)], may be obtained in an easier way. Another advantage of the technique presented in this paper is that it may also be used in order to construct symmetric \(n\)-noids.
Reviewer: Luc Vrancken (Valenciennes)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A35 | Non-Euclidean differential geometry |
| 49Q10 | Optimization of shapes other than minimal surfaces |
Citations:
Zbl 1144.53017References:
| [1] | I. Biswas, On the existence of unitary flat connections over the punctured sphere with given local monodromy around the punctures, Asian J. Math 3 (1999), 333-344. Zbl0982.14022 MR1796505 · Zbl 0982.14022 |
| [2] | A. I. Bobenko, Constant mean curvature surfaces and integrable equations, Russian Math. Surveys 46 (1991), 1-45. Zbl0780.53009 MR1138951 · Zbl 0780.53009 · doi:10.1070/RM1991v046n04ABEH002826 |
| [3] | J. Dorfmeister and G. Haak, Construction of non-simply connected CMC surfaces via dressing, J. Math. Soc. Japan 55 (2003), 335-364. Zbl1035.53015 MR1961290 · Zbl 1035.53015 · doi:10.2969/jmsj/1191419120 |
| [4] | J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633-668. Zbl0932.58018 MR1664887 · Zbl 0932.58018 |
| [5] | J. Dorfmeister and H. Wu, Construction of constant mean curvature trinoids from holomorphic potentials, preprint, 2000. Zbl1151.53055 · Zbl 1151.53055 · doi:10.1007/s00209-007-0197-1 |
| [6] | J. Dorfmeister and H. WuUnitarization of loop group representations of fundamental groups, preprint, 2005. Zbl1135.22020 MR2354553 · Zbl 1135.22020 |
| [7] | W. M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), 557-607. Zbl0655.57019 MR952283 · Zbl 0655.57019 · doi:10.1007/BF01410200 |
| [8] | K. Große-Brauckmann, New surfaces of constant mean curvature, Math. Z. 214 (1993), 527-565. Zbl0806.53005 MR1248112 · Zbl 0806.53005 · doi:10.1007/BF02572424 |
| [9] | K. Große-Brauckmann, R. Kusner and J. M. Sullivan, Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero, J. Reine Angew. Math. 564 (2003), 35-61. Zbl1058.53005 MR2021033 · Zbl 1058.53005 · doi:10.1515/crll.2003.093 |
| [10] | K. Große-Brauckmann, R. Kusner and J. M. Sullivan, Coplanar constant mean curvature surfaces, math.DG/0509210, 2005. Zbl1145.53002 · Zbl 1145.53002 · doi:10.4310/CAG.2007.v15.n5.a4 |
| [11] | N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three space, Ann. of Math. 131 (1990), 239-330. Zbl0699.53007 MR1043269 · Zbl 0699.53007 · doi:10.2307/1971494 |
| [12] | M. Kilian, W. Rossman and N. Schmitt, Delaunay ends of constant mean curvature surfaces, preprint, 2006. Zbl1144.53015 MR2388561 · Zbl 1144.53015 · doi:10.1112/S0010437X07003119 |
| [13] | M. Kilian, N. Schmitt and I. Sterling, Dressing CMC n-noids, Math. Z. 246 (2004), 501-519. Zbl1065.53010 MR2073454 · Zbl 1065.53010 · doi:10.1007/s00209-003-0587-y |
| [14] | N. Korevaar, R. Kusner, W. Meeks III and B. Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), 1-43. Zbl0757.53032 MR1147718 · Zbl 0757.53032 · doi:10.2307/2374738 |
| [15] | N. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465-503. Zbl0726.53007 MR1010168 · Zbl 0726.53007 |
| [16] | R. Mazzeo and F. Pacard, Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 (2001), 169-237. Zbl1005.53006 MR1807955 · Zbl 1005.53006 |
| [17] | I. McIntosh, Global solutions of the elliptic 2d periodic Toda lattice, Nonlinearity 7 (1994), 85-108. Zbl0840.58023 MR1260134 · Zbl 0840.58023 · doi:10.1088/0951-7715/7/1/005 |
| [18] | A. Pressley and G. Segal, “Loop Groups”, Oxford Science Monographs, Oxford Science Publications, 1988. Zbl0638.22009 · Zbl 0638.22009 |
| [19] | J. Ratzkin, An end to end gluing construction for metrics of constant positive scalar curvature, Indiana Univ. Math. J. 52 (2003), 703-726. Zbl1032.53028 MR1986894 · Zbl 1032.53028 · doi:10.1512/iumj.2003.52.2281 |
| [20] | N. Schmitt, CMCLab, http://www.gang.umass.edu/software. |
| [21] | N. Schmitt, Constant mean curvature \(n\)-noids with symmetries, preprint, 2006. |
| [22] | N. Schmitt, M. Kilian, S. Kobayashi and W. Rossman, Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms, J. London Math. Society, to appear. Zbl1144.53017 · Zbl 1144.53017 · doi:10.1112/jlms/jdm005 |
| [23] | M. Umehara and K. Yamada, Metrics of constant curvature 1 with three conical singularities on the 2-sphere, Illinois. J. Math. 44 (2000), 72-94. Zbl0958.30029 MR1731382 · Zbl 0958.30029 |
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