Equiboundedness of the Weil-Petersson metric

Wolpert, Scott

doi:10.1090/tran/6998

Equiboundedness of the Weil-Petersson metric
HTML articles powered by AMS MathViewer

by Scott A. Wolpert PDF

Trans. Amer. Math. Soc. 369 (2017), 5871-5887 Request permission

Abstract:

Given a topological type for surfaces of negative Euler characteristic, uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $C^0$, $C^{2,\alpha }$ and $L^2$ norms for harmonic Beltrami differentials are analyzed. Uniform bounds are given for the covariant derivatives of the Weil-Petersson curvature tensor in terms of the systoles of the underlying Riemann surfaces and the projections of the differentiation directions onto pinching directions. The main analysis combines Schauder and potential theory estimates with the analytic implicit function theorem.

References

Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
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
L. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975–978. MR 148896, DOI 10.1090/S0002-9939-1962-0148896-1
Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. MR 297992, DOI 10.2307/1970638
Melvin S. Berger, Nonlinearity and functional analysis, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR 0488101
Keith Burns, Howard Masur, Carlos Matheus, and Amie Wilkinson, Rates of mixing for the Weil-Petersson geodesic flow I: No rapid mixing in non-exceptional moduli spaces, Adv. Math. 306 (2017), 589–602. MR 3581311, DOI 10.1016/j.aim.2016.10.026
Keith Burns, Howard Masur, Carlos Matheus, and Amie Wilkinson, Rates of mixing for the Weil-Petersson geodesic flow: exponential mixing in exceptional moduli spaces, Geom. Funct. Anal. 27 (2017), no. 2, 240–288. MR 3626613, DOI 10.1007/s00039-017-0401-3
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
S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), 179–195. MR 19983, DOI 10.1090/S0002-9904-1947-08778-4
Clifford J. Earle, Some remarks on the Beltrami equation, Math. Scand. 36 (1975), 44–48. MR 374417, DOI 10.7146/math.scand.a-11561
John D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293(294) (1977), 143–203. MR 506038, DOI 10.1515/crll.1977.293-294.143
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
Zheng Huang, On asymptotic Weil-Petersson geometry of Teichmüller space of Riemann surfaces, Asian J. Math. 11 (2007), no. 3, 459–484. MR 2372726, DOI 10.4310/AJM.2007.v11.n3.a5
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
Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Canonical metrics on the moduli space of Riemann surfaces. I, J. Differential Geom. 68 (2004), no. 3, 571–637. MR 2144543
H. E. Rauch, Weierstrass points, branch points, and moduli of Riemann surfaces, Comm. Pure Appl. Math. 12 (1959), 543–560. MR 110798, DOI 10.1002/cpa.3160120310
Stefano Trapani, On the determinant of the bundle of meromorphic quadratic differentials on the Deligne-Mumford compactification of the moduli space of Riemann surfaces, Math. Ann. 293 (1992), no. 4, 681–705. MR 1176026, DOI 10.1007/BF01444740
Scott Wolpert, The Fenchel-Nielsen deformation, Ann. of Math. (2) 115 (1982), no. 3, 501–528. MR 657237, DOI 10.2307/2007011
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, The Bers embeddings and the Weil-Petersson metric, Duke Math. J. 60 (1990), no. 2, 497–508. MR 1047763, DOI 10.1215/S0012-7094-90-06020-X
Scott A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), no. 2, 417–472. MR 1037410
Scott A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67–89, 91–129. MR 1156387, DOI 10.1007/BF02100600
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, Geodesic-length functions and the Weil-Petersson curvature tensor, J. Differential Geom. 91 (2012), no. 2, 321–359. MR 2971291