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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 $C_2$.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 76-81
• Publisher Identifier: S 1079-6762(99)00064-5
• 1991 Mathematics Subject Classification. Primary 58G99
• Key words and phrases.
• Received by the editors March 25, 1999
• Posted on June 28, 1999
• Communicated by Walter Neumann
• Comments (When Available)

Robert Brooks

Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Israel

E-mail address: rbrooks@tx.technion.ac.il

Eran Makover

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

E-mail address: eranm@math.huji.ac.il

Current address: Department of Mathematics, Dartmouth College, Hanover, NH

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.