The first eigenvalue of a Riemann surface

Authors:
Robert Brooks and Eran Makover

Journal:
Electron. Res. Announc. Amer. Math. Soc. **5** (1999), 76-81

MSC (1991):
Primary 58G99

DOI:
https://doi.org/10.1090/S1079-6762-99-00064-5

Published electronically:
June 28, 1999

MathSciNet review:
1696823

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a collection of results whose central theme is that the phenomenon of the first eigenvalue of the Laplacian being large is typical for Riemann surfaces. Our main analytic tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under compactification of the surface. We make the notion of picking a Riemann surface at random by modeling this process on the process of picking a random $3$-regular graph. With this model, we show that there are positive constants $C_1$ and $C_2$ independent of the genus, such that with probability at least $C_1$, a randomly picked surface has first eigenvalue at least $C_2$.

- L. Ahlfors,
*An extension of Schwarz’ lemma*, Trans. AMS 43 (1938), 359–364. - G. V. Belyĭ,
*Galois extensions of a maximal cyclotomic field*, Izv. Akad. Nauk SSSR Ser. Mat.**43**(1979), no. 2, 267–276, 479 (Russian). MR**534593** - Béla Bollobás,
*Random graphs*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR**809996** - Béla Bollobás,
*The isoperimetric number of random regular graphs*, European J. Combin.**9**(1988), no. 3, 241–244. MR**947025**, DOI https://doi.org/10.1016/S0195-6698%2888%2980014-3 - R. Brooks,
*Some geometric aspects of the work of Lars Ahlfors*, to appear in Proc. Ahlfors Mem. Lect. - R. Brooks,
*Platonic surfaces*, Comm. Math. Helv. 74 (1999), 156–170. - Robert Brooks,
*The spectral geometry of a tower of coverings*, J. Differential Geom.**23**(1986), no. 1, 97–107. MR**840402** - R. Brooks,
*Twist surfaces*, to appear in Proc. Cortona Conf. - Robert Brooks,
*Some remarks on volume and diameter of Riemannian manifolds*, J. Differential Geom.**27**(1988), no. 1, 81–86. MR**918458** - R. Brooks and E. Makover,
*Riemann surfaces with large first eigenvalue*, to appear. - R. Brooks and E. Makover,
*The spectral geometry of Belyi surfaces*, to appear in Isr. Math. Conf. Proc. - R. Brooks and E. Makover,
*Random construction of Riemann surfaces*, to appear. - Peter Buser,
*Cubic graphs and the first eigenvalue of a Riemann surface*, Math. Z.**162**(1978), no. 1, 87–99. MR**505920**, DOI https://doi.org/10.1007/BF01437826 - Peter Buser,
*On the bipartition of graphs*, Discrete Appl. Math.**9**(1984), no. 1, 105–109. MR**754431**, DOI https://doi.org/10.1016/0166-218X%2884%2990093-3 - 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 https://doi.org/10.1007/BFb0083046 - 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** - W. Luo, Z. Rudnick, and P. Sarnak,
*On Selberg’s eigenvalue conjecture*, Geom. Funct. Anal.**5**(1995), no. 2, 387–401. MR**1334872**, DOI https://doi.org/10.1007/BF01895672 - Atle Selberg,
*On the estimation of Fourier coefficients of modular forms*, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1–15. MR**0182610** - Carsten Thomassen,
*Bidirectional retracting-free double tracings and upper embeddability of graphs*, J. Combin. Theory Ser. B**50**(1990), no. 2, 198–207. MR**1081223**, DOI https://doi.org/10.1016/0095-8956%2890%2990074-A - Nguyen Huy Xuong,
*Sur les immersions d’un graphe dans les surfaces orientables*, C. R. Acad. Sci. Paris Sér. A-B**283**(1976), no. 10, Ai, A745–A747. MR**440555** - Nguyen Huy Xuong,
*Sur quelques classes de graphes possédant des propriétés topologiques remarquables*, C. R. Acad. Sci. Paris Sér. A-B**283**(1976), no. 11, Ai, A813–A816 (French, with English summary). MR**424601**

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (1991):
58G99

Retrieve articles in all journals with MSC (1991): 58G99

Additional Information

**Robert Brooks**

Affiliation:
Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel

Email:
rbrooks@tx.technion.ac.il

**Eran Makover**

Affiliation:
Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311

Address at time of publication:
Department of Mathematics, Dartmouth College, Hanover, NH

Email:
eranm@math.huji.ac.il

Received by editor(s):
March 25, 1999

Published electronically:
June 28, 1999

Additional Notes:
Partially supported by the Israel Science Foundation, founded by the Israel Academy of Arts and Sciences, the Fund for the Promotion of Research at the Technion, and the New York Metropolitan Fund.

Communicated by:
Walter Neumann

Article copyright:
© Copyright 1999
American Mathematical Society