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On the Weil-Petersson metric on Teichmüller space

Authors: A. E. Fischer and A. J. Tromba
Journal: Trans. Amer. Math. Soc. 284 (1984), 319-335
MSC: Primary 32G15; Secondary 58B20
MathSciNet review: 742427
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Abstract: Teichmüller space for a compact oriented surface $ M$ without boundary is described as the quotient $ \mathcal{A}/{\mathcal{D}_0}$, where $ \mathcal{A}$ is the space of almost complex structures on $ M$ (compatible with a given orientation) and $ {\mathcal{D}_0}$ are those $ {C^\infty }$ diffeomorphisms homotopic to the identity. There is a natural $ {\mathcal{D}_0}$ invariant $ {L_2}$ Riemannian structure on $ \mathcal{A}$ which induces a Riemannian structure on $ \mathcal{A}/{\mathcal{D}_0}$. Infinitesimally this is the bilinear pairing suggested by Andre Weil--the Weil-Petersson Riemannian structure. The structure is shown to be Kähler with respect to a naturally induced complex structure on $ \mathcal{A}/{\mathcal{D}_0}$.

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