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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 application to infinitely dimensional Teichmtiller spaces.
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Lipman Bers (1960) studied this question.
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