On the Weil-Petersson convex geometry of Teichmüller space
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Sumio Yamada
Translated by: Sumio Yamada - Sugaku Expositions 30 (2017), 159-186
- DOI: https://doi.org/10.1090/suga/422
- Published electronically: September 15, 2017
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Abstract:
The family of conformal structures that can be realized on a given topological surface constitute the Teichmüller space of the surface. It is known to be simply connected and finite dimensional when the surface is of finite type. We will equip the Teichmüller space with a particular Riemannian metric, the Weil-Petersson metric, and investigate the convex geometry induced from the resulting distance function. In particular, we illustrate that the Weil-Petersson geometry offers a synthetic view of the so-called augmented Teichmüller space, through the theory of the CAT(0) space. Additionally, we will consider the Weil-Petersson metric defined on the universal Teichmüller space, where each finite-dimensional Teichmüller space is isometrically embedded. In conclusion, we formulate a question concerning the synthetic Weil-Petersson geometry of the universal Teichmüller space.References
- William Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44. MR 442293, DOI 10.2307/1971024
- A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360
- Lars V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math. (2) 74 (1961), 171–191. MR 204641, DOI 10.2307/1970309
- S. I. Al′ber, Spaces of mappings into a manifold of negative curvature, Dokl. Akad. Nauk SSSR 178 (1968), 13–16 (Russian). MR 0230254
- Abdelhadi Belkhirat, Athanase Papadopoulos, and Marc Troyanov, Thurston’s weak metric on the Teichmüller space of the torus, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3311–3324. MR 2135749, DOI 10.1090/S0002-9947-05-03735-9
- Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 43–55. MR 0361051
- N. Bourbaki, Groupes et Algébres de Lie, Hermann, Paris, 1981.
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- K. Burns, H. Masur, and A. Wilkinson, The Weil-Petersson geodesic flow is ergodic, Ann. of Math. (2) 175 (2012), no. 2, 835–908. MR 2993753, DOI 10.4007/annals.2012.175.2.8
- Tienchen Chu, The Weil-Petersson metric in the moduli space, Chinese J. Math. 4 (1976), no. 2, 29–51. MR 590105
- Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
- Georgios Daskalopoulos and Richard Wentworth, Classification of Weil-Petersson isometries, Amer. J. Math. 125 (2003), no. 4, 941–975. MR 1993745
- Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999
- J. Eells and L. Lemaire, Deformations of metrics and associated harmonic maps, Geometry and analysis, Indian Acad. Sci., Bangalore, 1980, pp. 33–45. MR 592252
- James Eells and Luc Lemaire, Two reports on harmonic maps, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1363513, DOI 10.1142/9789812832030
- 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
- Arthur E. Fischer and Jerrold E. Marsden, Deformations of the scalar curvature, Duke Math. J. 42 (1975), no. 3, 519–547. MR 380907
- A. E. Fischer and A. J. Tromba, On the Weil-Petersson metric on Teichmüller space, Trans. Amer. Math. Soc. 284 (1984), no. 1, 319–335. MR 742427, DOI 10.1090/S0002-9947-1984-0742427-X
- Arthur E. Fischer and Anthony J. Tromba, On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann. 267 (1984), no. 3, 311–345. MR 738256, DOI 10.1007/BF01456093
- Paul Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind, Math. Ann. 101 (1929), no. 1, 226–237 (German). MR 1512527, DOI 10.1007/BF01454835
- Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. MR 1215595
- Philip Hartman, On homotopic harmonic maps, Canadian J. Math. 19 (1967), 673–687. MR 214004, DOI 10.4153/CJM-1967-062-6
- Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
- Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
- Jürgen Jost and Richard Schoen, On the existence of harmonic diffeomorphisms, Invent. Math. 66 (1982), no. 2, 353–359. MR 656629, DOI 10.1007/BF01389400
- 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
- Norihito Koiso, Variation of harmonic mapping caused by a deformation of Riemannian metric, Hokkaido Math. J. 8 (1979), no. 2, 199–213. MR 551551, DOI 10.14492/hokmj/1381758271
- N. Korevaar and R. Schoen, Global existence theorems for harmonic maps: finite rank spaces and an approach to rigidity for smooth actions, Preprint (1997).
- 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
- Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
- Maryam Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), no. 1, 1–23. MR 2257394, DOI 10.1090/S0894-0347-06-00526-1
- Charles B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. MR 1501936, DOI 10.1090/S0002-9947-1938-1501936-8
- Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
- Subhashis Nag and Alberto Verjovsky, $\textrm {Diff}(S^1)$ and the Teichmüller spaces, Comm. Math. Phys. 130 (1990), no. 1, 123–138. MR 1055689
- Kunio Obitsu and Scott A. Wolpert, Grafting hyperbolic metrics and Eisenstein series, Math. Ann. 341 (2008), no. 3, 685–706. MR 2399166, DOI 10.1007/s00208-008-0210-y
- Athanase Papadopoulos and Marc Troyanov, Weak Finsler structures and the Funk weak metric, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 419–437. MR 2525935, DOI 10.1017/S0305004109002461
- Yu. G. Rešetnyak, On the theory of spaces with curvature no greater than $K$, Mat. Sb. (N.S.) 52 (94) (1960), 789–798 (Russian). MR 0121762
- Gonzalo Riera, A formula for the Weil-Petersson product of quadratic differentials, J. Anal. Math. 95 (2005), 105–120. MR 2145560, DOI 10.1007/BF02791498
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
- O. Teichmüller, Extremal quasiconformal mappings and quadratic differentials [translation of MR0003242]. Translated from the German by Guillaume Théret. IRMA Lect. Math. Theor. Phys., 26, Handbook of Teichmüller theory. Vol. V, 321-483, Eur. Math. Soc., Zürich, 2016.
- W. Thurston, Minimal stretch maps between hyperbolic surfaces. Preprint, available at arxiv:math GT/9801039 (1986).
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
- Anthony J. Tromba, Teichmüller theory in Riemannian geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. Lecture notes prepared by Jochen Denzler. MR 1164870, DOI 10.1007/978-3-0348-8613-0
- Leon A. Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc. 183 (2006), no. 861, viii+119. MR 2251887, DOI 10.1090/memo/0861
- L. A. Takhtajan and P. G. Zograf, The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces, J. Geom. Phys. 5 (1988), no. 4, 551–570 (1989). MR 1075722, DOI 10.1016/0393-0440(88)90019-8
- André Weil, Modules des surfaces de Riemann, Séminaire Bourbaki; 10e année: 1957/1958., Secrétariat mathématique, Paris, 1958, pp. 7 (French). Textes des conférences; Exposés 152 à 168; 2e éd. corrigée, Exposé 168. MR 0124485
- André Weil, Scientific works. Collected papers. Vol. II (1951–1964), Springer-Verlag, New York-Heidelberg, 1979 (French). MR 537935
- Scott Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), no. 2, 573–577. MR 422692
- Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. MR 842050, DOI 10.1007/BF01388794
- Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
- Scott A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 357–393. MR 2039996, DOI 10.4310/SDG.2003.v8.n1.a13
- Scott A. Wolpert, Behavior of geodesic-length functions on Teichmüller space, J. Differential Geom. 79 (2008), no. 2, 277–334. MR 2420020
- Scott A. Wolpert, Families of Riemann surfaces and Weil-Petersson geometry, CBMS Regional Conference Series in Mathematics, vol. 113, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2010. MR 2641916, DOI 10.1090/cbms/113
- Sumio Yamada, Weil-Peterson [Petersson] convexity of the energy functional on classical and universal Teichmüller spaces, J. Differential Geom. 51 (1999), no. 1, 35–96. MR 1703604
- S. Yamada, Weil-Petersson Completion of Teichmueller Spaces and Mapping Class Group Actions. Preprint arXiv:math/0112001v1 (2001).
- Sumio Yamada, On the geometry of Weil-Petersson completion of Teichmüller spaces, Math. Res. Lett. 11 (2004), no. 2-3, 327–344. MR 2067477, DOI 10.4310/MRL.2004.v11.n3.a5
- Sumio Yamada, Weil-Petersson geometry of Teichmüller-Coxeter complex and its finite rank property, Geom. Dedicata 145 (2010), 43–63. MR 2600944, DOI 10.1007/s10711-009-9401-2
- Sumio Yamada, Some aspects of Weil-Petersson geometry of Teichmüller spaces, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 531–546. MR 2906941
- Sumio Yamada, Local and global aspects of Weil-Petersson geometry, Handbook of Teichmüller theory. Vol. IV, IRMA Lect. Math. Theor. Phys., vol. 19, Eur. Math. Soc., Zürich, 2014, pp. 43–111. MR 3289699, DOI 10.4171/117-1/2
- Sumio Yamada, Convex bodies in Euclidean and Weil-Petersson geometries, Proc. Amer. Math. Soc. 142 (2014), no. 2, 603–616. MR 3134001, DOI 10.1090/S0002-9939-2013-11841-1
Bibliographic Information
- Sumio Yamada
- Affiliation: Department of Mathematics, Gakushuin University, Tokyo 171-8588, Japan
- MR Author ID: 641808
- Email: yamada@math.gakushuin.ac.jp
- Published electronically: September 15, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Sugaku Expositions 30 (2017), 159-186
- MSC (2010): Primary 20F55, 30F60, 32G15, 35J88, 51M10, 58J60
- DOI: https://doi.org/10.1090/suga/422
- MathSciNet review: 3711763