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Annals of Global Analysis and Geometry

, Volume 55, Issue 3, pp 479–507 | Cite as

Holomorphic quadratic differentials dual to Fenchel–Nielsen coordinates

  • Nadine GroßeEmail author
  • Melanie Rupflin
Article

Abstract

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel–Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The precise estimates derived in the current paper form the basis for the proof of the sharp eigenvalue estimates on degenerating surfaces obtained by the authors in another paper.

Keywords

Geometric analysis Functions on degenerating hyperbolic surfaces Fenchel–Nielsen coordinates 

References

  1. 1.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Progress in Mathematics, vol. 106. Birkhäuser, Boston (1992)zbMATHGoogle Scholar
  2. 2.
    Große, N., Rupflin, M.: Sharp eigenvalue estimates on degenerating surfaces (2017). arXiv:1701.08491
  3. 3.
    Hummel, C.: Gromov’s Compactness Theorem for Pseudo-holomorphic Curves. Progress in Mathematics, vol. 151. Birkhäuser, Basel (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Masur, H.: Extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43, 623–635 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mazzeo, R., Swoboda, J.: Asymptotics of the Weil-Petersson metric. IMRN 6, 1749–1786 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Randol, B.: Cylinders in Riemann surfaces. Comment. Math. Helv. 54, 1–5 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rupflin, M., Topping, P.M.: A uniform Poincaré estimate for quadratic differentials on closed surfaces. Calc. Var. Partial Differential Equations 53, 587–604 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rupflin, M., Topping, P.M.: Teichmüller harmonic map flow into nonpositively curved targets. J. Differential Geometry 108(1), 135–184 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rupflin, M., Topping, P.M., Zhu, M.: Asymptotics of the Teichmüller harmonic map flow. Adv. Math. 244, 874–893 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rupflin, M., Topping, P.M.: Horizontal curves of hyperbolic metrics. Calc. Var. 57, 106 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wolpert, S.: The Fenchel–Nielsen deformation. Ann. of Math. 3, 501–528 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wolpert, S.: Spectral limits for hyperbolic surfaces. II. Invent. Math. 108(1), 91–129 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wolpert, S.: Geometry of the Weil–Petersson completion of Teichmüller space. Surv. Differ. Geom. VIII, 357–393 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Wolpert, S.: Behavior of geodesic-length functions on Teichmüller space. J. Differential Geom. 79, 277–334 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wolpert, S.: Geodesic-length functions and the Weil–Petersson curvature tensor. J. Differential Geom. 91, 321–359 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yamada, S.: On the Weil–Petersson geometry of Teichmüller spaces. Math. Res. Lett. 11(3), 327–344 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yamada, S.: Local and Global Aspects of Weil–Petersson Geometry. Handbook of Teichmüller Theory IV. EMS, Zurich (2014). arXiv:1206.2083v2 Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität FreiburgFreiburgGermany
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK

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