## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.math.psu.edu/era/.
**

The first eigenvalue of a Riemann surface
**This journal is archived by the American Mathematical
Society. The master copy is available at
http://www.ams.org/era/
**

## The first eigenvalue of a Riemann surface

### Robert Brooks and Eran Makover

**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 $C_2$.

*Copyright 1999 American Mathematical Society*

**Retrieve entire article **

#### 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.

*Electronic Research Announcements of the AMS *Home page