**Contemporary Mathematics**

Volume: 201;
1997;
243 pp;
Softcover

MSC: Primary 30;
Secondary 14; 58

Print ISBN: 978-0-8218-0514-5

Product Code: CONM/201

List Price: $63.00

Individual Member Price: $50.40

**Electronic ISBN: 978-0-8218-7792-0
Product Code: CONM/201.E**

List Price: $63.00

Individual Member Price: $50.40

# Extremal Riemann Surfaces

Share this page *Edited by *
*J. R. Quine; Peter Sarnak*

This volume is an outgrowth of the AMS Special Session on
Extremal Riemann Surfaces held at the Joint Mathematics Meeting in San
Francisco, January 1995. The book deals with a variety of extremal
problems related to Riemann surfaces. Some papers deal with the
identification of surfaces with longest systole (element of shortest
nonzero length) for the length spectrum and the Jacobian. Parallels
are drawn to classical questions involving extremal lattices. Other
papers deal with maximizing or minimizing functions defined by the
spectrum such as the heat kernel, the zeta function, and the
determinant of the Laplacian, some from the point of view of
identifying an extremal metric.

There are discussions of Hurwitz surfaces and surfaces with large
cyclic groups of automorphisms. Also discussed are surfaces which are
natural candidates for solving extremal problems such as triangular,
modular, and arithmetic surfaces, and curves in various group
theoretically defined curve families. Other allied topics are theta
identities, quadratic periods of Abelian differentials,
Teichmüller disks, binary quadratic forms, and spectral
asymptotics of degenerating hyperbolic three manifolds.

Features:

- Includes papers by some of the foremost experts on Riemann surfaces.
- Outlines interesting connections between Riemann surfaces and parallel fields.
- Follows up on investigations of Sarnak concerning connections between the theory of extreme lattices and Jacobians of Riemann surfaces.
- Contains papers on a variety of topics relating to Riemann surfaces.

#### Table of Contents

# Table of Contents

## Extremal Riemann Surfaces

- Contents vii8 free
- Preface ix10 free
- Extremal geometries 112 free
- Extremal Riemann surfaces with a large number of systoles 920
- On arithmetic genus 2 subgroups of triangle groups 2132
- Some lattices obtained from Riemann surfaces 2940
- Jacobian of the Picard curve 3344
- Fermat's quartic curve, Klein's curve, and the tetrahedron 4354
- Riemann surfaces admitting large automorphism groups 6374
- The splitting of some Jacobi varieties using their automorphism groups 8192
- Number theory, theta identities, and modular curves 125136
- Uniformization of some quotients of modular curves 155166
- Teichmüller disks and Veech's F-structures 165176
- On the geometry and spectral asymptotics of degenerating hyperbolic three manifolds 191202
- Differential equations for the quadratic periods of abelian differentials 207218
- Zeta functions on S2 213224
- A minimum problem for heat kernels of flat tori 227238

#### Readership

Graduate students, research mathematicians interested in Riemannian surfaces and related fields.