Small eigenvalues of closed Riemann surfaces for large genus

Wu, Yunhui; Xue, Yuhao

doi:10.1090/tran/8608

Small eigenvalues of closed Riemann surfaces for large genus
HTML articles powered by AMS MathViewer

by Yunhui Wu and Yuhao Xue PDF

Trans. Amer. Math. Soc. 375 (2022), 3641-3663 Request permission

Abstract:

In this article we study the asymptotic behavior of small eigenvalues of hyperbolic surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the minimum of $k$-th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac {1}{g^2}$ in $g$. And the minimum of $ag$-th eigenvalues of hyperbolic surfaces in any thick part of moduli space is bounded above by a uniform constant only depending on $\varepsilon$ and $a$.

In the proof of the upper bound, for any constant $\varepsilon >0$, we will construct a closed hyperbolic surface of genus $g$ in any $\varepsilon$-thick part of moduli space such that it admits a pants decomposition whose curves all have length equal to $\varepsilon$, and the number of separating systole curves in this surface is uniformly comparable to $g$.

References

Werner Ballmann, Henrik Matthiesen, and Sugata Mondal, Small eigenvalues of closed surfaces, J. Differential Geom. 103 (2016), no. 1, 1–13. MR 3488128
Béla Bollobás, The asymptotic number of unlabelled regular graphs, J. London Math. Soc. (2) 26 (1982), no. 2, 201–206. MR 675164, DOI 10.1112/jlms/s2-26.2.201
Jeffrey F. Brock and Kenneth W. Bromberg, Inflexibility, Weil-Peterson distance, and volumes of fibered 3-manifolds, Math. Res. Lett. 23 (2016), no. 3, 649–674. MR 3533189, DOI 10.4310/MRL.2016.v23.n3.a4
Robert Brooks and Eran Makover, Riemann surfaces with large first eigenvalue, J. Anal. Math. 83 (2001), 243–258. MR 1828493, DOI 10.1007/BF02790263
P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus, Invent. Math. 117 (1994), no. 1, 27–56. With an appendix by J. H. Conway and N. J. A. Sloane. MR 1269424, DOI 10.1007/BF01232233
Peter Buser, Riemannsche Flächen mit Eigenwerten in $(0,$ $1/4)$, Comment. Math. Helv. 52 (1977), no. 1, 25–34. MR 434961, DOI 10.1007/BF02567355
Peter Buser, On Cheeger’s inequality $\lambda _{1}\geq h^{2}/4$, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 29–77. MR 573428
Peter Buser, On the bipartition of graphs, Discrete Appl. Math. 9 (1984), no. 1, 105–109. MR 754431, DOI 10.1016/0166-218X(84)90093-3
Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
Peter Buser, Marc Burger, and Jozef Dodziuk, Riemann surfaces of large genus and large $\lambda _1$, Geometry and analysis on manifolds (Katata/Kyoto, 1987) Lecture Notes in Math., vol. 1339, Springer, Berlin, 1988, pp. 54–63. MR 961472, DOI 10.1007/BFb0083046
William Cavendish and Hugo Parlier, Growth of the Weil-Petersson diameter of moduli space, Duke Math. J. 161 (2012), no. 1, 139–171. MR 2872556, DOI 10.1215/00127094-1507312
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289–297. MR 378001, DOI 10.1007/BF01214381
Paul Erdős and Horst Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 12 (1963), 251–257 (German). MR 165515
Alastair Fletcher, Jeremy Kahn, and Vladimir Markovic, The moduli space of Riemann surfaces of large genus, Geom. Funct. Anal. 23 (2013), no. 3, 867–887. MR 3061775, DOI 10.1007/s00039-013-0211-1
M. Fortier Bourque and K. Rafi, Local maxima of the systole function, J. Eur. Math. Soc. (2021), To appear.
Will Hide and Michael Magee, Near optimal spectral gaps for hyperbolic surfaces, Preprint, arXiv:2107.05292, July 2021.
Linda Keen, Collars on Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 263–268. MR 0379833
Michael Lipnowski and Alex Wright, Towards optimal spectral gaps in large genus, Preprint, arXiv:2103.07496, March 2021.
Tatiana Mantuano, Discretization of compact Riemannian manifolds applied to the spectrum of Laplacian, Ann. Global Anal. Geom. 27 (2005), no. 1, 33–46. MR 2130531, DOI 10.1007/s10455-005-5215-0
Maryam Mirzakhani, On Weil-Petersson volumes and geometry of random hyperbolic surfaces, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 1126–1145. MR 2827834
Maryam Mirzakhani, Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus, J. Differential Geom. 94 (2013), no. 2, 267–300. MR 3080483
Sugata Mondal, On largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces, Math. Z. 281 (2015), no. 1-2, 333–348. MR 3384873, DOI 10.1007/s00209-015-1486-8
David Mumford, A remark on Mahler’s compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294. MR 276410, DOI 10.1090/S0002-9939-1971-0276410-4
Jean-Pierre Otal and Eulalio Rosas, Pour toute surface hyperbolique de genre $g,\ \lambda _{2g-2}>1/4$, Duke Math. J. 150 (2009), no. 1, 101–115 (French, with English and French summaries). MR 2560109, DOI 10.1215/00127094-2009-048
Bram Petri, Hyperbolic surfaces with long systoles that form a pants decomposition, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1069–1081. MR 3750219, DOI 10.1090/proc/13806
Kasra Rafi and Jing Tao, The diameter of the thick part of moduli space and simultaneous Whitehead moves, Duke Math. J. 162 (2013), no. 10, 1833–1876. MR 3079261, DOI 10.1215/00127094-2323128
Peter Sarnak, Selberg’s eigenvalue conjecture, Notices Amer. Math. Soc. 42 (1995), no. 11, 1272–1277. MR 1355461
R. Schoen, S. Wolpert, and S. T. Yau, Geometric bounds on the low eigenvalues of a compact surface, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 279–285. MR 573440
Richard Schoen, A lower bound for the first eigenvalue of a negatively curved manifold, J. Differential Geometry 17 (1982), no. 2, 233–238. MR 664495
Nicholas C. Wormald, The asymptotic distribution of short cycles in random regular graphs, J. Combin. Theory Ser. B 31 (1981), no. 2, 168–182. MR 630980, DOI 10.1016/S0095-8956(81)80022-6
Yunhui Wu, Growth of the Weil-Petersson inradius of moduli space, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 3, 1309–1346 (English, with English and French summaries). MR 3986917, DOI 10.5802/aif.3272
Yunhui Wu and Yuhao Xue, Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus, Amer. J. Math. (2021), To appear.
Yunhui Wu and Yuhao Xue, Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac {3}{16}-\epsilon$, Geometric and Functional Analysis, (2022), to appear.