A uniform Poincaré estimate for quadratic differentials on closed surfaces

  • Melanie Rupflin
  • Peter Topping


We revisit the classical Poincaré inequality on closed surfaces, and prove its natural analogue for quadratic differentials. In stark contrast to the classical case, our inequality does not degenerate when we work on hyperbolic surfaces that themselves are degenerating, and this fact turns out to be essential for applications to the Teichmüller harmonic map flow.

Mathematics Subject Classification 

30F10 30F30 30F45 30F60 32G15 35A23 35J46 53A30 58J05 



Partially supported by The Leverhulme Trust and EPSRC Grant number EP/K00865X/1.


  1. 1.
    Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Hummel, C.: Gromov’s Compactness Theorem for Pseudo-Holomorphic Curves. Progress in Mathematics, vol. 151, p. 131. Birkhäuser Verlag, Basel (1997)CrossRefGoogle Scholar
  3. 3.
    Mumford, D.: A remark on Mahler’s compactness theorem. Proc. Am. Math. Soc. 28, 289–294 (1971)Google Scholar
  4. 4.
    Randol, B.: Cylinders in Riemann surfaces. Comment. Math. Helvetici 54, 1–5 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Rupflin, M., Topping, P.M.: Flowing maps to minimal surfaces. (2012)
  6. 6.
    Rupflin, M., Topping, P.M.: Teichmüller harmonic map flow into nonpositively curved targets. (2014)
  7. 7.
    Rupflin, M., Topping, P.M., Zhu, M.: Asymptotics of the Teichmüller harmonic map flow. Adv. Math. 244, 874–893 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Topping, P.M.: Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow. Ann. Math. 159, 465–534 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Tromba, A.: Teichmüller Theory in Riemannian Geometry. Lecture Notes Prepared by Jochen Denzler. Lectures in Mathematics ETH-Zürich. Birkhäuser, Boston (1992)CrossRefGoogle Scholar
  10. 10.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Progress in Mathematics, vol. 106. Birkhäuser Boston Inc., Boston (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of LeipzigLeipzigGermany
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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