Donald Bers

Mathema Tics: L. Bers

On Approximation in the Bers Spaces

Let D be a Jordan domain in the complex plane with rectifiable boundary C. Let Aq(D) denote the Bers space with norm || ||,. We prove that if/6 A„(D), 2<q<cc, then there exist functions ■s„W = 2£=i l/(z — zn!c), znltEC for k = \,■ ■ ,n, such that \Un— f\\«—"-0This result does not hold for 1 <q^2 even when D is a disc.

On the Bers Fiber Spaces

It is well known that a Bers fiber space F (Γ) for a torsion free finitely generated Fuchsian group Γ of the first kind can be identified with a Teichmüller space. If Γ has torsion, a theorem of Earle–Kra [9] asserts that in almost all cases, F (Γ) is not isomorphic to any Teichmüller space. However, there are 39 cases which remain unsettled. Our results remove 27 from the 39 previously unknown...

Bers Slices Are Zariski Dense

Each Bers slice is a holomorphically embedded copy of Teichmüller space within XC(S). While it follows that BY can be locally described as the common zero locus of finitely many analytic functions on XC(S), it is known that the Bers slice is not a locally algebraic set [DK]—this is used to show that W. Thurston’s skinning map is not a constant function [DK]. We prove a stronger result about the...

Simultaneous Uniformization by Lipman Bers

We shall show that any two Riemann surfaces satisfying a certain condition, for instance, any two closed surfaces of the same genus g > l , can be uniformized by one group of fractional linear transformations (Theorem 1). This leads, in conjunction with previous results [2; 3] , to the simultaneous uniformization of all algebraic curves of a given genus (Theorems 2-4). Theorem 5 contains an app...