Weil–Petersson metric

In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space T_g,n of genus g Riemann surfaces with n marked points. It was introduced by André Weil (1958, 1979) using the Petersson inner product on forms on a Riemann surface (introduced by Hans Petersson).

Definition

If a point of Teichmüller space is represented by a Riemann surface R, then the cotangent space at that point can be identified with the space of quadratic differentials at R. Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric.

Properties

Weil (1958) stated, and Ahlfors (1961) proved, that the Weil–Petersson metric is a Kähler metric. Ahlfors (1961b) proved that it has negative holomorphic sectional, scalar, and Ricci curvatures. The Weil–Petersson metric is usually not complete.

Generalizations

The Weil–Petersson metric can be defined in a similar way for some moduli spaces of higher-dimensional varieties.

References

Ahlfors, Lars V. (1961), "Some remarks on Teichmüller's space of Riemann surfaces", Annals of Mathematics. Second Series, 74: 171–191, JSTOR 1970309, MR 0204641
Ahlfors, Lars V. (1961b), "Curvature properties of Teichmüller's space", Journal d'Analyse Mathématique, 9: 161–176, doi:10.1007/BF02795342, MR 0136730
Weil, André (1958), "Modules des surfaces de Riemann", Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152à 168; 2e éd.corrigée, Exposé 168 (in French), Paris: Secrétariat Mathématique, pp. 413–419, MR 0124485, Zbl 0084.28102
Weil, André (1979) [1958], "On the moduli of Riemann surfaces", Scientific works. Collected papers. Vol. II (1951--1964), Berlin, New York: Springer-Verlag, pp. 381–389, ISBN 978-0-387-90330-9, MR 537935
Wolpert, Scott A. (2001), "Weil–Petersson_metric", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Wolpert, Scott A. (2009), "The Weil-Petersson metric geometry", in Papadopoulos, Athanase, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, pp. 47–64, arXiv:0801.0175, doi:10.4171/055-1/2, MR 2497791
Wolpert, Scott A. (2010), Families of Riemann Surfaces and Weil-Petersson Geometry, CBMS Reg. Conf. Series in Math. (113), Amer. Math. Soc., Providence, Rhode Island, ISBN 978-0-8218-4986-6, MR 2641916

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