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, Teichmuller disks, binary quadratic forms, and spectral asymptotics of degenerating hyperbolic three manifolds.

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. Gonzálvez-Aguilera -- 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 -- Teichmuller 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

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