In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space Tg,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 edit
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 edit
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 edit
The Weil–Petersson metric can be defined in a similar way for some moduli spaces of higher-dimensional varieties.
See also edit
References edit
- Ahlfors, Lars V. (1961), "Some remarks on Teichmüller's space of Riemann surfaces", Annals of Mathematics, Second Series, 74 (1): 171–191, doi:10.2307/1970309, hdl:2027/mdp.39015095258003, 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, hdl:2027/mdp.39015095248350, MR 0136730, S2CID 124921349
- 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 0537935
- Wolpert, Scott A. (2001) [1994], "Weil–Petersson_metric", Encyclopedia of Mathematics, EMS Press
- Wolpert, Scott A. (2009), "The Weil-Petersson metric geometry", in Papadopoulos, Athanase (ed.), Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, pp. 47–64, arXiv:0801.0175, doi:10.4171/055-1/2, ISBN 978-3-03719-055-5, MR 2497791
- Wolpert, Scott A. (2010), Families of Riemann Surfaces and Weil-Petersson Geometry, CBMS Reg. Conf. Series in Math., vol. 113, Amer. Math. Soc., Providence, Rhode Island, arXiv:1202.4078, doi:10.1090/cbms/113, ISBN 978-0-8218-4986-6, MR 2641916, S2CID 7880175