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On largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces

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Abstract

We apply topological methods to study the smallest non-zero number \(\lambda _1\) in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set \(\{ S \in {{\mathcal {M}}_2}: {\lambda _1}(S) > \frac{1}{4} \}\) is unbounded and disconnects the moduli space \({{\mathcal {M}}_2}\). Using this, for genus \(g \ge 3\), we show the existence of eigenbranches that start as \(\lambda _1\) and eventually becomes \({>} \frac{1}{4}\).

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References

  1. Bers, L.: A remark on Mumford’s compactness theorem. Isr. J. Math. 12, 400–407 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buser, P.: Cubic graphs and the first eigenvalue of a Riemann surface. Math. Z. 162, 87–99 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  3. Buser, P.: On the bipartition of graphs. Discrete Appl. Math. 9, 105–109 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Progress in Mathematics, vol. 106. Birkhäuser Boston Inc, Boston (1992)

    Google Scholar 

  5. Burger, M., Buser, P., Dodziuk, J.: Riemann surfaces of large genus and large \(\lambda _1\). Geometry and analysis on manifolds. In: Sunada, T. (ed.) Lecture Notes in Mathematics, vol. 1339, pp. 54–63. Springer, Berlin (1988)

    Google Scholar 

  6. Brooks, R., Makover, E.: Riemann surfaces with large first eigenvalue. J. Anal. Math. 83, 243–258 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115. Academic Press, London (1984)

    Google Scholar 

  8. Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51, 43–55 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Colbois, B., Courtois, G.: Les valeurs propres inférieures á 1/4 des surfaces de Riemann de petit rayon d’injectivité. Comment. Math. Helv. 64(3), 349–362 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Colbois, B., de Verdire, Y.C.: Sur la multiplicit de la premire valeur propre d’une surface de Riemann courbure constante (French) (Multiplicity of the first eigenvalue of a Riemann surface with constant curvature). Comment. Math. Helv. 63(2), 194–208 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hatcher, A.: Algebraic Topology (English summary), xii+544 pp. Cambridge University Press, Cambridge (2002). ISBN: 0-521-79160-X

  12. Hejhal, D.: Regular b-groups, degenerating Riemann surfaces and spectral theory. Mem. Am. Math. Soc 88, 437 (1990)

    MathSciNet  Google Scholar 

  13. Huxley, M.N.: Cheeger’s inequality with a boundary term. Comment. Math. Helv. 58, 347–354 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. Iwaniec, H.: Introduction to the Spectral Theory of Automorphic Forms. Bibl. Rev. Mat. Iberoamericana, Revista Matemática Iberoamericana, Madrid (1995)

  15. Jenni, F.: Uber den ersten Eigenwert des Laplace-Operators auf ausgewhlten Beispielen kompakter Riemannscher Flchen (German) [On the first eigenvalue of the Laplace operator on selected examples of compact Riemann surfaces]. Comment. Math. Helv. 59(2), 193–203 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ji, L.: Spectral degeneration of hyperbolic Riemann surfaces. J. Differ. Geom. 38(2), 263–313 (1993)

    MATH  Google Scholar 

  17. Judge, C.: The nodal set of a finite sum of Maass cusp forms is a graph. In: Proceedings of Symposia in Pure Mathematics, vol. 84 (2012)

  18. Kim, H.H.: Functoriality for the exterior square of \(GL_4\) and symmetric fourth of \(GL_2\). J. Am. Math. Soc. 16(1), 139–183 (2003)

    Article  MATH  Google Scholar 

  19. Mondal, S.: Topological bounds on the number of cuspidal eigenvalues of finite area hyperbolic surfaces. Int. Math. Res. Not. (to appear)

  20. Otal, J.-P.: Three topological properties of small eigenfunctions on hyperbolic surfaces. In: Geometry and Dynamics of Groups and Spaces, Progr. Math., vol. 265. Birkhäuser, Bassel (2008)

  21. Otal, J.-P., Rosas, E.: Pour toute surface hyperbolique de genre g, \({\lambda _{2g-2}}>1/4\). Duke Math. J. 150(1), 101–115 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Proceedings of the Symposium on Pure Mathematics VII, Am. Math. Soc., pp. 1–15 (1965)

  23. Strohmaier, A., Uski, V.: An algorithm for the computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces. Commun. Math. Phys. 317(3), 827–869 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wolpert, S.A.: Spectral limits for hyperbolic surface, I. Invent. Math. 108, 67–89 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I would like to thank my advisor Jean-Pierre Otal for all his help starting from suggesting the problem to me. I am thankful to Peter Buser and Werner Ballmann for the discussions that I had with them on this problem. Finally I would like to thank the Max Planck Institute for Mathematics in Bonn for its support and hospitality.

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Correspondence to Sugata Mondal.

Appendix

Appendix

For the convenience of the reader we give a proof of the fact that, for \((g, n) \ne (0, 4), (1, 1)\), the complement \({{\mathcal {M}}_{g, n}} {\setminus } {\mathcal {I}}_\epsilon \) of the compact set \({\mathcal {I}}_\epsilon = \{S \in {{\mathcal {M}}_{g, n}}: s(S) \ge \epsilon \}\) [1] is path connected.

Lemma 5.3

For any \((g, n) \ne (0, 4), (1, 1)\) with \(2g-2+n > 0\) and any \(\epsilon >0\) the set \({{\mathcal {M}}_{g, n}} {\setminus } {\mathcal {I}}_\epsilon \) is path connected.

Proof

Let \(S_1\) and \(S_2\) be two surfaces in \({{\mathcal {M}}_{g, n}}\) such that \(s(S_i) < \epsilon \). So we have simple closed geodesics \(\gamma _1\) on \(S_1\) and \(\gamma _2\) on \(S_2\) such that the length \(l_{\gamma _i}\) of \(\gamma _i\) is \(< \epsilon \). Recall that it has always been our practise to treat \({\mathcal {M}}_{g, n}\) as a subset of all possible metrics on a fixed surface S and the geodesics are understood to be parametric curves on S that satisfy certain differential equations provided by the metric.

With this understanding let us first assume that \(\gamma _1\) does not intersect \(\gamma _2\). So we may consider a pants decomposition P of S containing both \(\gamma _1\) and \(\gamma _2\). Let the Fenchel–Nielsen coordinates of \(S_i\) be given by \(({l_j}(S_i), {\theta _j}(S_i))_{j=1}^{3g-3+n}\). Here \(l_1\), \(l_2\) are the length parameters along \(\gamma _1, \gamma _2\) and \(\theta _1, \theta _2\) are twist parameters along \(\gamma _1, \gamma _2\). Then consider the path \(\beta : [0, 1] \rightarrow {\mathcal {T}}_2\) given by:

$$\begin{aligned} {l_1}(\beta (t))= & {} {\left\{ \begin{array}{ll} {l_1}(S_1)\,&{}\quad \text {if }t \in [0, \frac{1}{2}], \\ 2(1-t){l_1}(S_1) +(2t-1){l_1}(S_2) &{}\quad \text {if }t \in [\frac{1}{2}, 1] \end{array}\right. }\\ {l_2}(\beta (t))= & {} {\left\{ \begin{array}{ll} (1-2t){l_2}(S_1) + 2t{l_2}(S_2)\,&{}\quad \text {if }t \in [0, \frac{1}{2}], \\ {l_2}(S_2) &{}\quad \text {if }t \in [\frac{1}{2}, 1] \end{array}\right. } \end{aligned}$$

\({l_3}(\beta (t))=(1-t){l_3}(S_1) + t {l_3}(S_2)\) and \({\theta _j}(\beta (t))=(1-t){\theta _j}(S_1) + t {\theta _j}(S_2)\). Since \(l_1(\beta (t)) < \epsilon \) for \(t \in [0, \frac{1}{2}]\) and \(l_2(\beta (t)) < \epsilon \) for \(t \in [\frac{1}{2}, 1]\) we observe that \(s(\beta (t)) < \epsilon \) for all t. The image of \(\beta \) under the quotient map \({\mathcal {T}}_{g, n} \rightarrow {\mathcal {M}}_{g, n}\) produces the required path joining \(S_1\) and \(S_2\).

Now let us assume that \(\gamma _1\) intersects \(\gamma _2\). Let \(\gamma \) be a simple closed geodesic that does not intersect \(\gamma _1\) and \(\gamma _2\). By our assumption i.e. \((g, n) \ne (0, 4), (1, 1)\) such a geodesic exists. Then by the procedure described above both \(S_1\) and \(S_2\) can be joined by a path in \({{\mathcal {M}}_{g, n}} {\setminus } {\mathcal {I}}_\epsilon \) to a surface on which \(\gamma \) has length \(< \epsilon \). This finishes the proof.

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Mondal, S. On largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces. Math. Z. 281, 333–348 (2015). https://doi.org/10.1007/s00209-015-1486-8

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