Contemporary Mathematics 1997; 243 pp; softcover Volume: 201 ISBN10: 0821805142 ISBN13: 9780821805145 List Price: US$60 Member Price: US$48 Order Code: CONM/201
 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.
Readership Graduate students, research mathematicians interested in Riemannian surfaces and related fields. Table of Contents  P. Sarnak  Extremal geometries
 P. S. Schaller  Extremal Riemann surfaces with a large number of systoles
 N. Näätänen and T. Kuusalo  On arithmetic genus 2 subgroups of triangle groups
 M. Bernstein and N. A. Sloane  Some lattices obtained from Riemann surfaces
 J. R. Quine  Jacobian of the Picard curve
 R. E. Rodríguez and V. GonzálezAguilera  Fermat's quartic curve, Klein's curve, and the tetrahedron
 J. X. Ries  The splitting of some Jacobi varieties using their automorphism groups
 R. S. Kulkarni  Riemann surfaces admitting large automorphism gorups
 R. Brooks, H. M. Farkas, and I. Kra  Number theory, theta identities, and modular curves
 R. Brooks and Y. Kopeliovich  Uniformization of some quotients of modular curves
 C. J. Earle and F. P. Gardiner  Teichmüller disks and Veech's \(\mathcal F\)structures
 J. Dodziuk and J. Jorgenson  On the geometry and spectral asymptotics of degenerating hyperbolic three manifolds
 D. Curtis and M. Tretkoff  Differential equations for the quadratic periods of abelian differentials
 C. Morpurgo  Zeta functions on \(S^2\)
 A. B. II  A minimum problem for heat kernels of flat tori
