Harmonic maps into hyperbolic $3$-manifolds
HTML articles powered by AMS MathViewer
- by Yair N. Minsky PDF
- Trans. Amer. Math. Soc. 332 (1992), 607-632 Request permission
Abstract:
High-energy degeneration of harmonic maps of Riemann surfaces into a hyperbolic $3$-manifold target is studied, generalizing results of [M1] in which the target is two-dimensional. The Hopf foliation of a high-energy map is mapped to an approximation of its geodesic representative in the target, and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy. The images of harmonic maps obtained when the domain degenerates along a Teichmüller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable.References
- Robert Brooks and J. Peter Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982), no. 1, 163–182. MR 650375 F. Bonahon, Bouts des variétés hyperboliques de dimension $3$]], Ann. of Math. (2) 124 (1986), 71-158. D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc. (to appear).
- R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92. MR 903850
- James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510, DOI 10.1090/cbms/050
- 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
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- A. E. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30 (1988), no. 1, 63–88. MR 964063, DOI 10.1016/0166-8641(88)90081-8 J. Harer and R. Penner, Princeton Univ. Press, 1991.
- Jürgen Jost and Hermann Karcher, Almost linear functions and a priori estimates for harmonic maps, Global Riemannian geometry (Durham, 1983) Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1984, pp. 148–155. MR 757216
- Jürgen Jost and Hermann Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), no. 1, 27–77 (German, with English summary). MR 679120, DOI 10.1007/BF01168235
- Jürgen Jost, Harmonic maps between surfaces, Lecture Notes in Mathematics, vol. 1062, Springer-Verlag, Berlin, 1984. MR 754769, DOI 10.1007/BFb0100160
- Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23–41. MR 559474, DOI 10.1016/0040-9383(80)90029-4 Y. Minsky, Harmonic maps and hyperbolic geometry, Ph.D. thesis, Princeton University, 1989.
- Yair N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992), no. 1, 151–217. MR 1152229
- J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 211–228. MR 510549
- Richard M. Schoen, Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 321–358. MR 765241, DOI 10.1007/978-1-4612-1110-5_{1}7
- G. P. Scott, Compact submanifolds of $3$-manifolds, J. London Math. Soc. (2) 7 (1973), 246–250. MR 326737, DOI 10.1112/jlms/s2-7.2.246
- Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
- William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277 —, The geometry and topology of $3$-manifolds, Princeton University lecture notes, 1982. —, Minimal stretch maps between hyperbolic surfaces, unpublished manuscript, Princeton University.
- K. Uhlenbeck, Harmonic maps; a direct method in the calculus of variations, Bull. Amer. Math. Soc. 76 (1970), 1082–1087. MR 264714, DOI 10.1090/S0002-9904-1970-12570-8
- 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
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 332 (1992), 607-632
- MSC: Primary 58E20; Secondary 20H10, 30F60, 32G15, 32G34, 57M50, 57N10, 57R42
- DOI: https://doi.org/10.1090/S0002-9947-1992-1100698-9
- MathSciNet review: 1100698