Lipman Bers

An extremal problem for quasiconformal mappings and a theorem by Thurston

The new extremal problem considered in this paper, and the form of the solution are suggested by Thurstons beautiful theorem on the structure of topological self-mappings of a surface (Theorem 4 in [21]). In treating this problem, however, we use none of Thursforts results and thus obtain also a new proof of his theorem. Unless otherwise stated all surfaces considered in this paper will be assumed to be oriented and of finite type (p, m), tha t is homeomorphic to a sphere with p >/0 handles from which one has removed m ~>0 disjoint continua. To avoid uninteresting special cases we assume tha t 2 p 2 + m >0 . (1.1)

Finite dimensional Teichmüller spaces and generalizations

0. Background

Automorphic forms for Schottky groups

Publisher Summary The chapter discusses a (revised and expanded) version of a lecture delivered at the Los Alamos Workshop on Mathematics in June 1974 under the title A Glimpse into Complex Analysis. It discusses automorphic forms for Schottky groups. Schottky groups stand out, among other Kleinian groups, by a particularly simple algebraic and geometric structure. Every finitely generated free Kleinian group all of which nontrivial elements are loxodromic is a Schottky group. One can always construct standard fundamental regions bounded by analytic curves. There are Schottky groups for which no standard fundamental region, no matter what generators one chooses, is bounded by circles as every finitely generated subgroup of a Schottky group is a Schottky group.

Partial Differential Equations.

Many physical problems involve quantities that depend on more than one variable. The temperature within a “large”1 solid body of conducting material varies with both time and location within the material. When such problems are modeled, what results is a differential equation involving partial derivatives, or a partial differential equation..

An Inequality for Riemann Surfaces

This note contains a proof of an elementary but useful inequality for Riemann surfaces with a finitely generated non-Abelian fundamental group.

On rings of analytic functions

Let D be a domain in the complex plane (Riemann sphere) and R(D) the totality of one-valued regular analytic functions defined in D. With the usual definitions of addition and multiplication R(D) becomes a commutative ring (in fact, a domain of integrity). A oneto-one conformai transformation f =0(z) of D onto a domain A induces an isomorphism ƒ—>ƒ* between R(D) and R(A):f(z) =ƒ*[</>(2)]. An anti-conformal transformation

On spaces of Riemann surfaces with nodes

This is a summary of results, to be published in full elsewhere, which strengthen and refine the statements made in a previous announcement [1]. A compact Riemann surface with nodes of (arithmetic) genus p > 1 is a connected complex space S, on which there are k = k(S) > 0 points P j , • • • , Pk> called nodes, such that (i) every node P. has a neighborhood isomorphic to the analytic set {ZjZ2 = 0, \zt\ < 1, |z2 | < 1}, with Py corresponding to (0, 0); (ii) the set S\{Pl9 • • • , Pk] has r > 1 components S x , • • • , S r , called parts of S, each 2,. is a Riemann surface of some genus pi9 compact except for n( punctures, with 3pt 3 + nt > 0, and n1 + • • • + nr = 2k\ and (iii) we have

A remark on Mumford’s compactness theorem

It is shown that a recent compactness theorem for Fuchsian groups, due to Mumford, remains valid for groups containing elliptic and parabolic elements.

Automorphic Forms and General Teichmüller Spaces

This is a summary of results extending the theory of Teichmuller spaces [1, 2, 6, 7] to arbitrary Fuchsian groups and to open Riemann surfaces. The new point of view also sheds some light on the case of closed surfaces. Proofs will appear elsewhere.

Eichler integrals with singularities

Let 1 ~ be a non-elementary Kleinian group with region of discontinuity ~ and let q >~ 2 be an integer. We shall show that there exist Eichler integrals of degree q (this term, as all others used here, will be defined below) with preassigned singularities at finitely many non-equivalent points of ~ and with preassigned parabolic singularities at finitely many non-equivalent cusps, and that these integrals have certain pleasing properties. Our results are a modest improvement of those of Ahlfors [3], who constructed Eichler integrals with preassigned poles at preassigned ordinary points in ~. The method, however, may be of interest since it clarifies the connection between Eichler integrals with poles and generalized Beltrami coefficients (as defined in Bers [5]). That such a connection must exist becomes obvious, at least for a finitely generated group F, by comparing recent results of Ahlfors [3] with those of Kra [6].