Extension of the Weil-Petersson connection. (English) Zbl 1167.32010
Let \(\mathfrak{T}_{g,n}\) denote the space of marked Riemann surfaces of genus \(g\) with \(n\)-punctures. Weil-Petersson (WP) metric, the geodesic-length functions, and for \(n>0\) the functons with values the distance between unit-length horocycles are quantities associated to hyperbolic metrics on Riemann surfaces. The convexity of the length functions along WP-geodesics is a fundamental property. The author generalizes this property and proves that the distance between horocycles is convex along WP geodesics, too. In particulary, the author proves for surfaces with punctures that the distance between horocycles and WP geodesics are suitably approximated by geodesic lengths and WP geodesics for surfaces without punctures.
Reviewer: Yaşar Sözen (Istanbul)
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 30F60 | Teichmüller theory for Riemann surfaces |
Keywords:
Teichmüller space of punctured Riemann surfaces; Weil-Petersson (WP) geodesics; Weil-Petersson-Levi-Civita connection; Pinched hyperbolic metricsReferences:
| [1] | W. Abikoff, Degenerating families of Riemann surfaces , Ann. of Math. (2) 105 (1977), 29–44. JSTOR: · Zbl 0347.32010 · doi:10.2307/1971024 |
| [2] | -, The Real Analytic Theory of Teichmüller Space , Lecture Notes in Math. 820 , Springer, Berlin, 1980. · Zbl 0452.32015 |
| [3] | L. V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces , Ann. of Math. (2) 74 (1961), 171–191. JSTOR: · Zbl 0146.30602 · doi:10.2307/1970309 |
| [4] | L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics , Ann. of Math. (2) 72 (1960), 385–404. JSTOR: · Zbl 0104.29902 · doi:10.2307/1970141 |
| [5] | L. Bers, “Spaces of degenerating Riemann surfaces” in Discontinuous Groups and Riemann Surfaces (College Park, Md., 1973) , Ann. of Math. Stud. 79 , Princeton Univ. Press, Princeton, 1974, 43–55. · Zbl 0294.32016 |
| [6] | J. F. Brock, The Weil-Petersson metric and volumes of, \(3\)-dimensional hyperbolic convex cores , J. Amer. Math. Soc. 16 (2003), 495–535. · Zbl 1059.30036 · doi:10.1090/S0894-0347-03-00424-7 |
| [7] | P. Buser, Geometry and Spectra of Compact Riemann Surfaces , Progr. Math. 106 , Birkhäuser, Boston, 1992. · Zbl 0770.53001 |
| [8] | C. Chabauty, Limite d’ensembles et géométrie des nombres , Bull. Soc. Math. France 78 (1950), 143–151. · Zbl 0039.04101 |
| [9] | E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations , McGraw-Hill, New York, 1955. · Zbl 0064.33002 |
| [10] | G. Daskalopoulos and R. Wentworth, Classification of Weil-Petersson isometries , Amer. J. Math. 125 (2003), 941–975. · Zbl 1043.32007 · doi:10.1353/ajm.2003.0023 |
| [11] | J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group , J. Reine Angew. Math. 293 /294 (1977), 143–203. · Zbl 0352.30012 · doi:10.1515/crll.1977.293-294.143 |
| [12] | F. P. Gardiner, Schiffer’s interior variation and quasiconformal mapping , Duke Math. J. 42 (1975), 371–380. · Zbl 0347.30017 · doi:10.1215/S0012-7094-75-04235-0 |
| [13] | William Harvey, “Chabauty spaces of discrete groups” in Discontinuous Groups and Riemann Surfaces (College Park, Md., 1973) , Ann. of Math. Stud. 79 , Princeton Univ. Press, Princeton, 1974, 239–246. · Zbl 0313.32028 |
| [14] | Z. Huang, On asymptotic Weil-Petersson geometry of Teichmüller s pace of Riemann surfaces, Asian J. Math. 11 (2007), 459–484. · Zbl 1137.53009 · doi:10.4310/AJM.2007.v11.n3.a5 |
| [15] | Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces , rev. ed., Springer, Tokyo, 1992. · Zbl 0754.30001 |
| [16] | H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space , Duke Math. J. 43 (1976), 623–635. · Zbl 0358.32017 · doi:10.1215/S0012-7094-76-04350-7 |
| [17] | H. A. Masur and Y. N. Minsky, Geometry of the complex of curves, II : Hierarchical structure , Geom. Funct. Anal. 10 (2000), 902–974. · Zbl 0972.32011 · doi:10.1007/PL00001643 |
| [18] | G. Mcshane and R. C. Penner, Stable curves and screens on fatgraphs , preprint,\arxiv0707.1468v2[math.GT] |
| [19] | G. Mondello, Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure , preprint,\arxivmath/0610698v3[math.DG] · Zbl 1165.53026 |
| [20] | S. Nag, The Complex Analytic Theory of Teichmüller Spaces , Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Interscience, 1988. · Zbl 0667.30040 |
| [21] | B. O’Neill, Semi-Riemannian Geometry , Pure Appl. Math. 103 , Academic Press, New York, 1983. |
| [22] | R. C. Penner, The decorated Teichmüller space of punctured surfaces , Comm. Math. Phys. 113 (1987), 299–339. · Zbl 0642.32012 · doi:10.1007/BF01223515 |
| [23] | -, “Cell decomposition and compactification of Riemann’s moduli space in decorated Teichmüller theory” in Woods Hole Mathematics , Ser. Knots Everything 34 , World Sci. Publ., Hackensack, N.J., 2004. 263–301. |
| [24] | A. J. Tromba, Teichmüller Theory in Riemannian Geometry , Lectures Math. ETH Zurich, Birkhäuser, Basel, 1992. · Zbl 0785.53001 |
| [25] | S. A. Wolpert, The Fenchel-Nielsen deformation , Ann. of Math. (2) 115 (1982), 501–528. JSTOR: · Zbl 0496.30039 · doi:10.2307/2007011 |
| [26] | -, Geodesic length functions and the Nielsen problem , J. Differential Geom. 25 (1987), 275–296. · Zbl 0616.53039 |
| [27] | -, The hyperbolic metric and the geometry of the universal curve , J. Differential Geom. 31 (1990), 417–472. · Zbl 0698.53002 |
| [28] | -, Spectral limits for hyperbolic surfaces. I , Invent. Math. 108 (1992), 67–89.; II , 91–129. · Zbl 0772.11016 · doi:10.1007/BF02100600 |
| [29] | -, “Geometry of the Weil-Petersson completion of Teichmüller space” in Surveys in Differential Geometry, Vol. VIII (Boston, 2002) , Surv. Differ. Geom. 8 , Int. Press, Somerville, Mass., 2003, 357–393. · Zbl 1049.32020 |
| [30] | -, Behavior of geodesic-length functions on Teichmüller space , J. Differential Geom. 79 (2008), 277–334. · Zbl 1147.30032 |
| [31] | S. Yamada, Weil-Petersson c ompletion of Teichmüller spaces and mapping class group actions, preprint,\arxivmath/0112001v1[math.DG] |
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