Holomorphic Differentials as Functions of Moduli
نویسنده
چکیده
The purpose of this note is to strengthen the results of [3 ] and to indicate a very brief derivation of some theorems announced without proof in [ l ; 3 ] . We begin by indicating a correction to [3]. Let Si and S2 be Riemann surfaces, ƒ an orientation preserving (orientation reversing) homeomorphism of bounded eccentricity of Si onto 52 and [ƒ] the homotopy class of/; then (Si, [ƒ], S2) is called an even (odd) coupled pair of Riemann surfaces. The definition of equivalence of such pairs given in [3 ] is imprecise and garbled by misprints. The correct definition reads: (Si, [ƒ], S2) and (Si , [ƒ'], S2') are called equivalent if there exist conformai homeomorphisms hi and fe with Ai(Si)=Si , A2(S2) =S2 ' and [fe/] = [f'hi] ; the two pairs are called strongly equivalent if S{ = S2 and there exists a conformai homeomorphism h with h(Si) = S i and [ƒ] = [ƒ'&]. If S0 is a Riemann surface, then the Teichmüller space T(So) can be thought of as the set of strong equivalence of even pairs (S, [ƒ], So) (and not of simple equivalence classes as stated in [3]). From now on we assume that S0 is a fixed closed Riemann surface of genus g> 1, and we write T instead of T(S0). T has a natural complex analytic structure and can be represented as a bounded domain, homeomorphic to a ball, in complex number space with coordinates (moduli) TI, • • • , r3fl_3 (cf. [ l ; 2]). Points of Twill be denoted by r. We may assume that S0 is given as the unit disc modulo a fixedpoint-free Fuchsian group, and that r = 0 corresponds to the pair (So, [identity], S0).
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