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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
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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Additional Information

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

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

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

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