Growth of the Weil–Petersson inradius of moduli space
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, p. 1309-1346

In this paper we study the systole function along Weil–Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil–Petersson metric. As an application, we study the growth of the Weil–Petersson inradius of moduli space of Riemann surfaces of genus g with n punctures as a function of g and n. We show that the Weil–Petersson inradius is comparable to lng with respect to g, and is comparable to 1 with respect to n.

Moreover, we also study the asymptotic behavior, as g goes to infinity, of the Weil–Petersson volumes of geodesic balls of finite radii in Teichmüller space. We show that they behave like o((1 g) (3-ϵ)g ) as g, where ϵ>0 is arbitrary.

Dans cet article, nous étudions la fonction systole le long des géodésiques de la métrique de Weil–Petersson. Nous montrons que la racine carrée de la systole est uniformément Lipschitz sur l’espace de Teichmüller muni de la métrique de Weil–Petersson. Comme application, nous étudions la croissance du rayon de la plus grande boule métrique inscrite dans l’espace des modules des surfaces de Riemann de genre g avec n piqûres en fonction de g et n. Nous montrons que ce rayon est comparable à lng par rapport à g, et comparable à 1 par rapport à n.

De plus, nous étudions aussi le comportement asymptotique, lorsque g tends vers l’infini, des volumes de Weil–Petersson des boules géodésiques de rayons finis dans l’espace Teichmüller. Nous montrons qu’ils se comportent comme o((1 g) (3-ϵ)g ) quand g, où ϵ>0 est arbitraire.

Received : 2017-05-02
Revised : 2018-03-15
Accepted : 2018-05-07
Published online : 2019-06-03
DOI : https://doi.org/10.5802/aif.3272
Classification:  32G15,  30F60
Keywords: The moduli space, Weil–Petersson metric, inradius, large genus, systole
@article{AIF_2019__69_3_1309_0,
     author = {Wu, Yunhui},
     title = {Growth of the Weil--Petersson inradius of moduli space},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     pages = {1309-1346},
     doi = {10.5802/aif.3272},
     zbl = {07067433},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_3_1309_0}
}
Growth of the Weil–Petersson inradius of moduli space. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1309-1346. doi : 10.5802/aif.3272. https://aif.centre-mersenne.org/item/AIF_2019__69_3_1309_0/

[1] Ahlfors, Lars V. Some remarks on Teichmüller’s space of Riemann surfaces, Ann. Math., Tome 74 (1961), pp. 171-191 | Article | Zbl 0146.30602

[2] Akrout, H. Singularités topologiques des systoles généralisées, Topology, Tome 42 (2003) no. 2, pp. 291-308 | Article | MR 1941437 | Zbl 1054.32006

[3] Balacheff, Florent; Makover, Eran; Parlier, Hugo Systole growth for finite area hyperbolic surfaces, Ann. Fac. Sci. Toulouse, Math., Tome 23 (2014) no. 1, pp. 175-180 | Article | MR 3204736 | Zbl 1295.30093

[4] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 319 (1999) | MR 1744486 | Zbl 0988.53001

[5] Brock, Jeffrey The Weil–Peterson metric and volumes of 3-dimensional hyperbolic convex cores, J. Am. Math. Soc., Tome 16 (2003) no. 3, pp. 495-535 | Article | Zbl 1059.30036

[6] Brock, Jeffrey; Bromberg, Kenneth W. Inflexibility, Weil–Peterson distance, and volumes of fibered 3-manifolds, Math. Res. Lett., Tome 23 (2016) no. 3, pp. 649-674 | Article | MR 3533189 | Zbl 1377.30037

[7] Brock, Jeffrey; Margalit, Dan Weil–Petersson isometries via the pants complex, Proc. Am. Math. Soc., Tome 135 (2007) no. 3, pp. 795-803 | Article | MR 2262875 | Zbl 1110.32004

[8] Brock, Jeffrey; Masur, Howard; Minsky, Yair Asymptotics of Weil–Peterson geodesic. I. Ending laminations, recurrence, and flows, Geom. Funct. Anal., Tome 19 (2010) no. 5, pp. 1229-1257 | Article | Zbl 1216.32007

[9] Brock, Jeffrey; Masur, Howard; Minsky, Yair Asymptotics of Weil–Peterson geodesics II: bounded geometry and unbounded entropy, Geom. Funct. Anal., Tome 21 (2011) no. 4, pp. 820-850 | Article | Zbl 1227.32018

[10] Brock, Jeffrey; Modami, Babak Recurrent Weil–Petersson geodesic rays with non-uniquely ergodic ending laminations, Geom. Topol., Tome 19 (2015) no. 6, pp. 3565-3601 | Article | MR 3447110 | Zbl 1332.30067

[11] Buser, Peter; Sarnak, Peter On the period matrix of a Riemann surface of large genus, Invent. Math., Tome 117 (1994) no. 1, pp. 27-56 (With an appendix by J. H. Conway and N. J. A. Sloane) | Article | MR 1269424 | Zbl 0814.14033

[12] Cavendish, William; Parlier, Hugo Growth of the Weil–Peterson diameter of moduli space, Duke Math. J., Tome 161 (2012) no. 1, pp. 139-171 | Article | Zbl 1244.32008

[13] Chu, Tienchen The Weil–Peterson metric in the moduli space, Chin. J. Math., Tome 4 (1976) no. 2, pp. 29-51 | MR 590105 | Zbl 0344.32006

[14] Daskalopoulos, Georgios; Wentworth, Richard Classification of Weil–Peterson isometries, Am. J. Math., Tome 125 (2003) no. 4, pp. 941-975 | Article | Zbl 1043.32007

[15] Fay, John D. Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math., Tome 293/294 (1977), pp. 143-203 | MR 506038 | Zbl 0352.30012

[16] Fletcher, Alastair; Kahn, Jeremy; Markovic, Vladimir The moduli space of Riemann surfaces of large genus, Geom. Funct. Anal., Tome 23 (2013) no. 3, pp. 867-887 | Article | MR 3061775 | Zbl 1284.32007

[17] Fujiwara, Koji Geometry of the Funk metric on Weil–Peterson spaces, Math. Z., Tome 274 (2013) no. 1-2, pp. 647-665 | Article | MR 3054348 | Zbl 1326.32024

[18] Gardiner, Frederick P. Schiffer’s interior variation and quasiconformal mapping, Duke Math. J., Tome 42 (1975), pp. 371-380 | Article | MR 382637 | Zbl 0347.30017

[19] Gardiner, Frederick P. A correspondence between laminations and quadratic differentials, Complex Variables, Theory Appl., Tome 6 (1986) no. 2-4, pp. 363-375 | Article | MR 871741 | Zbl 0604.30053

[20] Gendulphe, Matthieu The injectivity radius of hyperbolic surfaces and some Morse functions over moduli spaces (2015) (https://arxiv.org/abs/1510.02581 )

[21] Gromov, Misha Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, Modern Birkhäuser Classics (2007), xx+585 pages | MR 2307192 | Zbl 1113.53001

[22] Guth, Larry; Parlier, Hugo; Young, Robert Pants Decompositions of Random Surfaces, Geom. Funct. Anal., Tome 21 (2011) no. 5, pp. 1069-1090 | Article | MR 2846383 | Zbl 1242.32007

[23] Hamenstaedt, Ursula Teichmueller flow and Weil–Peterson flow (2015) (https://arxiv.org/abs/1505.01113 )

[24] Huang, Zheng On asymptotic Weil–Peterson geometry of Teichmüller space of Riemann surfaces, Asian J. Math., Tome 11 (2007) no. 3, pp. 459-484 | Article | MR 2372726 | Zbl 1137.53009

[25] Huang, Zheng The Weil–Peterson geometry on the thick part of the moduli space of Riemann surfaces, Proc. Am. Math. Soc., Tome 135 (2007) no. 10, pp. 3309-3316 | Article | MR 2322763 | Zbl 1125.30037

[26] Hubbard, John Hamal Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1: Teichmüller theory, Matrix Editions (2006), xx+459 pages | Zbl 1102.30001

[27] Imayoshi, Yoichi; Taniguchi, Masahiko An introduction to Teichmüller spaces, Springer (1992) (Translated and revised from the Japanese by the authors) | Article | Zbl 0754.30001

[28] Keen, Linda Collars on Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton University Press (Annals of Mathematics Studies) Tome 79 (1974), pp. 263-268 | MR 379833 | Zbl 0304.30014

[29] Kerckhoff, Steven P. The Nielsen realization problem, Ann. Math., Tome 117 (1983) no. 2, pp. 235-265 | MR 690845 | Zbl 0528.57008

[30] Liu, Kefeng; Sun, Xiaofeng; Yau, Shing-Tung Good geometry on the curve moduli, Publ. Res. Inst. Math. Sci., Tome 44 (2008) no. 2, pp. 699-724 | MR 2426362 | Zbl 1219.14012

[31] Liu, Kefeng; Xu, Hao Recursion formulae of higher Weil–Peterson volumes, Int. Math. Res. Not. (2009) no. 5, pp. 835-859 | Zbl 1186.14059

[32] Masur, Howard Extension of the Weil–Peterson metric to the boundary of Teichmuller space, Duke Math. J., Tome 43 (1976) no. 3, pp. 623-635 | Zbl 0358.32017

[33] Masur, Howard; Wolf, Michael The Weil–Peterson isometry group, Geom. Dedicata, Tome 93 (2002), pp. 177-190 | Article | Zbl 1014.32008

[34] Mirzakhani, Maryam Weil–Petersson volumes and intersection theory on the moduli space of curves, J. Am. Math. Soc., Tome 20 (2007) no. 1, pp. 1-23 | Article | MR 2257394 | Zbl 1120.32008

[35] Mirzakhani, Maryam On Weil–Petersson volumes and geometry of random hyperbolic surfaces, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency (2010), pp. 1126-1145 | MR 2827834 | Zbl 1239.32013

[36] Mirzakhani, Maryam Growth of Weil–Peterson volumes and random hyperbolic surfaces of large genus, J. Differ. Geom., Tome 94 (2013) no. 2, pp. 267-300 | Article | MR 3080483 | Zbl 1270.30014

[37] Mirzakhani, Maryam; Zograf, Peter Towards large genus asymptotics of intersection numbers on moduli spaces of curves, Geom. Funct. Anal., Tome 25 (2015) no. 4, pp. 1258-1289 | Article | MR 3385633 | Zbl 1327.14136

[38] Mumford, David A remark on Mahler’s compactness theorem, Proc. Am. Math. Soc., Tome 28 (1971), pp. 289-294 | MR 276410 | Zbl 0215.23202

[39] Penner, Robert C. Weil–Petersson volumes, J. Differ. Geom., Tome 35 (1992) no. 3, pp. 559-608 http://projecteuclid.org/euclid.jdg/1214448257 | Article | MR 1163449 | Zbl 0768.32016

[40] Rafi, Kasra; Tao, Jing The diameter of the thick part of moduli space and simultaneous Whitehead moves, Duke Math. J., Tome 162 (2013) no. 10, pp. 1833-1876 | Article | MR 3079261 | Zbl 1277.32013

[41] Riera, Gonzalo A formula for the Weil–Peterson product of quadratic differentials, J. Anal. Math., Tome 95 (2005), pp. 105-120 | Article | MR 2145560 | Zbl 1085.30041

[42] Schmutz, Paul Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal., Tome 3 (1993) no. 6, pp. 564-631 | Article | MR 1250756 | Zbl 0810.53034

[43] Schmutz, Paul Congruence subgroups and maximal Riemann surfaces, J. Geom. Anal., Tome 4 (1994) no. 2, pp. 207-218 | Article | MR 1277506 | Zbl 0804.32010

[44] Schumacher, Georg Harmonic maps of the moduli space of compact Riemann surfaces, Math. Ann., Tome 275 (1986) no. 3, pp. 455-466 | Article | MR 858289 | Zbl 0601.32025

[45] Schumacher, Georg; Trapani, Stefano Estimates of Weil–Peterson volumes via effective divisors, Commun. Math. Phys., Tome 222 (2001), pp. 1-7 | Article | Zbl 0988.32013

[46] Teo, Lee-Peng The Weil–Peterson geometry of the moduli space of Riemann surfaces, Proc. Am. Math. Soc., Tome 137 (2009) no. 2, pp. 541-552 | MR 2448574 | Zbl 1180.30050

[47] Tromba, Anthony J. On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Peterson metric, Manuscr. Math., Tome 56 (1986) no. 4, pp. 475-497 | Article | Zbl 0606.32014

[48] Wolf, Michael The Weil–Peterson Hessian of length on Teichmüller space, J. Differ. Geom., Tome 91 (2012) no. 1, pp. 129-169 | Article | Zbl 1254.30076

[49] Wolf, Michael; Wu, Yunhui Uniform Bounds for Weil–Peterson Curvatures, Proc. Lond. Math. Soc., Tome 117 (2018) no. 5, pp. 1041-1076 | Article | Zbl 06991333

[50] Wolpert, Scott A. Noncompleteness of the Weil–Petersson metric for Teichmüller space, Pac. J. Math., Tome 61 (1975) no. 2, pp. 573-577 | Article | Zbl 0327.32009

[51] Wolpert, Scott A. The length spectra as moduli for compact Riemann surfaces, Ann. Math., Tome 109 (1979) no. 2, pp. 323-351 | MR 528966 | Zbl 0441.30055

[52] Wolpert, Scott A. The Fenchel-Nielsen deformation, Ann. Math., Tome 115 (1982) no. 3, pp. 501-528 | MR 657237 | Zbl 0496.30039

[53] Wolpert, Scott A. Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math., Tome 85 (1986) no. 1, pp. 119-145 | Article | MR 842050 | Zbl 0595.32031

[54] Wolpert, Scott A. Geodesic length functions and the Nielsen problem, J. Differ. Geom., Tome 25 (1987) no. 2, pp. 275-296 | Article | MR 880186 | Zbl 0616.53039

[55] Wolpert, Scott A. Geometry of the Weil–Petersson completion of Teichmüller space, Lectures on geometry and topology, International Press. (Surveys in Differential Geometry) Tome 8 (2003), pp. 357-393 | Zbl 1049.32020

[56] Wolpert, Scott A. Weil–Petersson perspectives, Problems on mapping class groups and related topics, American Mathematical Society (Proceedings of Symposia in Pure Mathematics) Tome 74 (2006), pp. 269-282 | Article | MR 2264546 | Zbl 1259.32004

[57] Wolpert, Scott A. Behavior of geodesic-length functions on Teichmüller space, J. Differ. Geom., Tome 79 (2008) no. 2, pp. 277-334 | Article | MR 845703 | Zbl 1147.30032

[58] Wolpert, Scott A. Families of Riemann surfaces and Weil–Peterson geometry, American Mathematical Society, CBMS Regional Conference Series in Mathematics, Tome 113 (2010), viii+118 pages | Zbl 1198.30049

[59] Wu, Yunhui Iteration of mapping classes and limits of Weil–Peterson geodesics (2012) (preprint)

[60] Wu, Yunhui The Riemannian sectional curvature operator of the Weil–Peterson metric and its application, J. Differ. Geom., Tome 96 (2014) no. 3, pp. 507-530 | MR 3189463 | Zbl 1295.30103

[61] Wu, Yunhui On the Weil–Peterson curvature of the moduli space of Riemann surfaces of large genus, Int. Math. Res. Not. (2017) no. 4, pp. 1066-1102 | MR 3658160 | Zbl 1405.30047

[62] Yamada, Sumio On the geometry of Weil–Peterson completion of Teichmüller spaces, Math. Res. Lett., Tome 11 (2004) no. 2-3, pp. 327-344 | Article | Zbl 1060.32005

[63] Zograf, Peter On the large genus asymptotics of Weil–Peterson volumes (2008) (https://arxiv.org/abs/0812.0544 )