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
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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 $C_2$.
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- R. Brooks, Platonic surfaces, Comm. Math. Helv. 74 (1999), 156–170.
- R. Brooks, The spectral geometry of a tower of coverings, J. Diff. Geom. 23 (1986), 97–107.
- R. Brooks, Twist surfaces, to appear in Proc. Cortona Conf.
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- 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.
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- P. Buser, On the bipartition of graphs, Discrete Appl. Math. 9 (1984), 105–109.
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- J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in: Gunning (ed.), Problems in Analysis, Princeton University Press (1970), pp. 195–199.
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- A. Selberg, On the estimation of Fourier coefficients of modular forms, in: A. L. Whiteman (ed.), Theory of Numbers, Proc. Symp. Pure Math. 8 (1965), pp. 1–15.
- C. Thomassen, Bidirectional retracting-free double tracings and upper imbeddability, J. Comb. Th. B 50 (1990), 198–207.
- N. H. Xuong, Sur les immersions d’un graphe dans les surfaces orientables, C. R. Acad. Sci. Paris 283 (1976), 745–747.
- N. H. Xuong, Sur quelques classes des graphes possédant des propriétés remarquables, C. R. Acad. Sci. Paris 283 (1976), 813–816.
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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
