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The first eigenvalue of a Riemann surface
Author(s):
Robert
Brooks;
Eran
Makover
Journal:
Electron. Res. Announc. Amer. Math. Soc.
5
(1999),
76-81.
MSC (1991):
Primary 58G99
Posted:
June 28, 1999
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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  -regular graph. With this model, we show that there are positive constants  and  independent of the genus, such that with probability at least  , a randomly picked surface has first eigenvalue at least  .
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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
DOI:
10.1090/S1079-6762-99-00064-5
PII:
S 1079-6762(99)00064-5
Received by editor(s):
March 25, 1999
Posted:
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
Copyright of article:
Copyright
1999,
American Mathematical Society
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