Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures
-
Martin Bridgeman
Abstract
In this article we show that for every finite area hyperbolic surface X of type
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1564410
Funding statement: Martin Bridgeman’s research was supported by NSF grant DMS-1564410. Yunhui Wu is partially supported by China’s Recruitment Program of Global Experts.
Acknowledgements
The authors would like to thank Jeffrey Brock, Ken Bromberg and Michael Wolf for helpful conversations on this project.
References
[1] P. Buser, Geometry and spectra of compact Riemann surfaces, Mod. Birkhäuser Class., Birkhäuser, Boston 2010. 10.1007/978-0-8176-4992-0Suche in Google Scholar
[2] P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus, Invent. Math. 117 (1994), no. 1, 27–56. 10.1007/BF01232233Suche in Google Scholar
[3] Z. Huang, Asymptotic flatness of the Weil–Petersson metric on Teichmüller space, Geom. Dedicata 110 (2005), 81–102. 10.1007/s10711-003-0816-xSuche in Google Scholar
[4] Z. Huang, The Weil–Petersson geometry on the thick part of the moduli space of Riemann surfaces, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3309–3316. 10.1090/S0002-9939-07-08868-5Suche in Google Scholar
[5] Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo 1992. 10.1007/978-4-431-68174-8Suche in Google Scholar
[6] K. Liu, X. Sun and S.-T. Yau, Good geometry on the curve moduli, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 699–724. 10.2977/prims/1210167341Suche in Google Scholar
[7] M. Mirzakhani, Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus, J. Differential Geom. 94 (2013), no. 2, 267–300. 10.4310/jdg/1367438650Suche in Google Scholar
[8] G. Schumacher, Harmonic maps of the moduli space of compact Riemann surfaces, Math. Ann. 275 (1986), no. 3, 455–466. 10.1007/BF01458616Suche in Google Scholar
[9] L.-P. Teo, The Weil–Petersson geometry of the moduli space of Riemann surfaces, Proc. Amer. Math. Soc. 137 (2009), no. 2, 541–552. 10.1090/S0002-9939-08-09692-5Suche in Google Scholar
[10] A. J. Tromba, On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Petersson metric, Manuscripta Math. 56 (1986), no. 4, 475–497. 10.1007/BF01168506Suche in Google Scholar
[11] M. Wolf, The Weil–Petersson Hessian of length on Teichmüller space, J. Differential Geom. 91 (2012), no. 1, 129–169. 10.4310/jdg/1343133703Suche in Google Scholar
[12] M. Wolf and Y. Wu, Uniform bounds for Weil-Petersson curvatures, Proc. Lond. Math. Soc. (3) 117 (2018), no. 5, 1041–1076. 10.1112/plms.12152Suche in Google Scholar
[13] S. A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. 10.1007/BF01388794Suche in Google Scholar
[14] S. A. Wolpert, Behavior of geodesic-length functions on Teichmüller space, J. Differential Geom. 79 (2008), no. 2, 277–334. 10.4310/jdg/1211512642Suche in Google Scholar
[15] S. A. Wolpert, Families of Riemann surfaces and Weil–Petersson geometry, CBMS Reg. Conf. Ser. Math. 113, American Mathematical Society, Providence 2010. 10.1090/cbms/113Suche in Google Scholar
[16] S. A. Wolpert, Geodesic-length functions and the Weil–Petersson curvature tensor, J. Differential Geom. 91 (2012), no. 2, 321–359. 10.4310/jdg/1344430826Suche in Google Scholar
[17] S. A. Wolpert, Equiboundedness of the Weil–Petersson metric, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5871–5887. 10.1090/tran/6998Suche in Google Scholar
[18] Y. Wu, The Riemannian sectional curvature operator of the Weil–Petersson metric and its application, J. Differential Geom. 96 (2014), no. 3, 507–530. 10.4310/jdg/1395321848Suche in Google Scholar
[19] Y. Wu, On the Weil–Petersson curvature of the moduli space of Riemann surfaces of large genus, Int. Math. Res. Not. IMRN 2017 (2017), no. 4, 1066–1102. 10.1093/imrn/rnw053Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Eichler cohomology and zeros of polynomials associated to derivatives of L-functions
- Subgroups of elliptic elements of the Cremona group
- On derived equivalences and homological dimensions
- Area minimizing surfaces of bounded genus in metric spaces
- Half-space theorems for the Allen–Cahn equation and related problems
- Stability conditions on product varieties
- Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures
- Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries
- On birational boundedness of foliated surfaces
Artikel in diesem Heft
- Frontmatter
- Eichler cohomology and zeros of polynomials associated to derivatives of L-functions
- Subgroups of elliptic elements of the Cremona group
- On derived equivalences and homological dimensions
- Area minimizing surfaces of bounded genus in metric spaces
- Half-space theorems for the Allen–Cahn equation and related problems
- Stability conditions on product varieties
- Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures
- Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries
- On birational boundedness of foliated surfaces
