On a purely ?Riemannian? proof of the structure and dimension of the unramified moduli space of a compact Riemann surface

Fischer, Arthur E.; Tromba, Anthony J.

doi:10.1007/bf01456093

Cited by 73 publications

(61 citation statements)

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“…Let the space si of oriented complex structures on^#be defined by (0.2) si= [f^C*{Tx\Jt))\f2 = -I, and the natural orientation induced by/ is that of Jt). In [7] we proved Theorem (0.3). Let M be an oriented, compact connected C00 2-manifold without boundary of genus greater than one.…”

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confidence: 81%

“…Clearly y/r = 3/3, Thus, the actual space of Riemann moduli for a compact surface without boundary is the quotient of Teichmüller space by the action of a discrete group. Identifying 3w ith si/3, we have proven the following [7] Theorem (0.4). LetJtbe a C°° compact oriented surface without boundary of genus p, p > 1.…”

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confidence: 88%

“…Introduction and statement of main results. In [7] the authors, using only concepts from Riemannian geometry and global nonlinear analysis, develop an "a priori" approach to Teichmüller theory. Teichmiiller's theorem states (roughly) that the space of conformally inequivalent Riemann surfaces of genus p, p > 1 (with some topological restrictions) is homeomorphic to Euclidean R6^6 space.…”

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confidence: 99%

“…Let M be an oriented, compact connected C00 2-manifold without boundary of genus greater than one. Then si carries the structure of a Cx strong ILH (inverse limit of Hubert; see [7,11] where " ° " denotes the composition of (1,1) tensor fields.…”

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confidence: 99%

“…Let 3R_X be those metrics with scalar curvature negative one. If dim Jt = 2 and genus Jt > 1, 9W_1 is a nonempty strong ILH manifold [7]. Let P be the space of C°° positive functions on Jt.…”

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On the Weil-Petersson metric on Teichmüller space

Fischer¹,

Tromba²

1984

Trans. Amer. Math. Soc.

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Abstract. Teichmüller space for a compact oriented surface M without boundary is described as the quotient jí/B0, where s?is the space of almost complex structures on M (compatible with a given orientation) and 8)0 are those C°° diffeomorphisms homotopic to the identity. There is a natural 3>0 invariant L2 Riemannian structure on si which induces a Riemannian structure on si/3>n. Infinitesimally this is the bilinear pairing suggested by Andre Weil-the Weil-Petersson Riemannian structure. The structure is shown to be Kahler with respect to a naturally induced complex structure on¿//S0. In this work we show how, in the context of the approach to Teichmüller theory developed in [7 and 8], the Weil-Petersson metric arises naturally. We shall provide a simple and direct proof that the metric is Kahler.We now present a more detailed description of our results. We begin first, however, with a review of our basic approach to the development of a Teichmüller theory and of some of the results in [7 and 8].Let .J'be a C°° finite-dimensional manifold without boundary. Let T\(Jt) be the vector bundle over J( of tensors of type (1,1), and CX(TX(J?)) the space of Cx sections.

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confidence: 81%

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confidence: 88%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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