Limits of harmonic maps and crowned hyperbolic surfaces

Gupta, Subhojoy

doi:10.1090/tran/7777

Limits of harmonic maps and crowned hyperbolic surfaces
HTML articles powered by AMS MathViewer

by Subhojoy Gupta PDF

Trans. Amer. Math. Soc. 372 (2019), 7573-7596 Request permission

Abstract:

We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$ from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of the existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target of the same topological type.

References

S. I. Al’ber, On n-dimensional problems of the calculus of variations in the large, Sov. Math. Dokl. 6 (1964), 700–704.
Thomas K. K. Au, Luen-Fai Tam, and Tom Y. H. Wan, Hopf differentials and the images of harmonic maps, Comm. Anal. Geom. 10 (2002), no. 3, 515–573. MR 1912257, DOI 10.4310/CAG.2002.v10.n3.a4
Francis Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), no. 2, 233–297 (English, with English and French summaries). MR 1413855, DOI 10.5802/afst.829
Francis Bonahon, Geodesic laminations on surfaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 1–37. MR 1810534, DOI 10.1090/conm/269/04327
Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Subhojoy Gupta, Harmonic maps and wild Teichmüller spaces, to appear in Journal of Topology and Analysis, arXiv:1708.04780, 2017.
Subhojoy Gupta, Meromorphic quadratic differentials with half-plane structures, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 1, 305–347. MR 3186818, DOI 10.5186/aasfm.2014.3908
Subhojoy Gupta, Asymptoticity of grafting and Teichmüller rays II, Geom. Dedicata 176 (2015), 185–213. MR 3347577, DOI 10.1007/s10711-014-9963-5
Zheng-Chao Han, Remarks on the geometric behavior of harmonic maps between surfaces, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994) A K Peters, Wellesley, MA, 1996, pp. 57–66. MR 1417948
Philip Hartman, On homotopic harmonic maps, Canadian J. Math. 19 (1967), 673–687. MR 214004, DOI 10.4153/CJM-1967-062-6
N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, DOI 10.1112/plms/s3-55.1.59
John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
Zheng-Chao Han, Luen-Fai Tam, Andrejs Treibergs, and Tom Wan, Harmonic maps from the complex plane into surfaces with nonpositive curvature, Comm. Anal. Geom. 3 (1995), no. 1-2, 85–114. MR 1362649, DOI 10.4310/CAG.1995.v3.n1.a3
Andy C. Huang, Harmonic maps of punctured surfaces to the hyperbolic plane, preprint, arXiv:1605.07715, 2016.
Jürgen Jost, Compact Riemann surfaces, Universitext, Springer-Verlag, Berlin, 1997. An introduction to contemporary mathematics; Translated from the German manuscript by R. R. Simha. MR 1632873, DOI 10.1007/978-3-662-03446-0
Michael Kapovich, Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2009. Reprint of the 2001 edition. MR 2553578, DOI 10.1007/978-0-8176-4913-5
Michael Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31. MR 357739, DOI 10.1007/BF01236981
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Peter Li and Luen-Fai Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), no. 1, 1–46. MR 1109619, DOI 10.1007/BF01232256
Yair N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992), no. 1, 151–217. MR 1152229
Yair N. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996), no. 2, 249–286. MR 1390649, DOI 10.1215/S0012-7094-96-08310-6
R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 211–228. MR 510549, DOI 10.24033/asens.1344
Jan Swoboda, Moduli spaces of Higgs bundles on degenerating Riemann surfaces, Adv. Math. 322 (2017), 637–681. MR 3720806, DOI 10.1016/j.aim.2017.10.028
Richard Schoen and Shing Tung Yau, On univalent harmonic maps between surfaces, Invent. Math. 44 (1978), no. 3, 265–278. MR 478219, DOI 10.1007/BF01403164
Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
Michael Wolf, High energy degeneration of harmonic maps between surfaces and rays in Teichmüller space, Topology 30 (1991), no. 4, 517–540. MR 1133870, DOI 10.1016/0040-9383(91)90037-5
Michael Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991), no. 2, 487–539. MR 1094467
Michael Wolf, Harmonic maps from a surface and degeneration in Teichmüller space, Low-dimensional topology (Knoxville, TN, 1992) Conf. Proc. Lecture Notes Geom. Topology, III, Int. Press, Cambridge, MA, 1994, pp. 217–239. MR 1316183, DOI 10.1159/000246841