Geodesic
The first eigenvalue of a Riemann surface
Brooks, Robert1 ; Makover, Eran2, 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
Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 76-81.

Voir la notice de l'article provenant de la source American Mathematical Society

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. http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-99-00064-5/

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