Geodesic
The first eigenvalue of a Riemann surface
Brooks, Robert1 ; Makover, Eran2, 3
1Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel
2Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311
3Department of Mathematics, Dartmouth College, Hanover, NH
Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 76-81
Cet article a éte moissonné depuis la source American Mathematical Society

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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 $C_2$.
DOI : 10.1090/S1079-6762-99-00064-5

Brooks, Robert  1   ; Makover, Eran  2 , 3

1 Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel
2 Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311
3 Department of Mathematics, Dartmouth College, Hanover, NH 
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Brooks, Robert; Makover, Eran. The first eigenvalue of a Riemann surface. Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 76-81. doi: 10.1090/S1079-6762-99-00064-5

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