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
editIf 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
editWeil (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
editThe Weil–Petersson metric can be defined in a similar way for some moduli spaces of higher-dimensional varieties.
See also
editReferences
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 0204641171-191&rft.date=1961&rft_id=info:hdl/2027/mdp.39015095258003&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0204641#id-name=MR&rft_id=https://www.jstor.org/stable/1970309#id-name=JSTOR&rft_id=info:doi/10.2307/1970309&rft.aulast=Ahlfors&rft.aufirst=Lars V.&rfr_id=info:sid/en.wikipedia.org:Weil–Petersson metric" class="Z3988">
- 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 124921349161-176&rft.date=1961&rft_id=info:hdl/2027/mdp.39015095248350&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0136730#id-name=MR&rft_id=https://api.semanticscholar.org/CorpusID:124921349#id-name=S2CID&rft_id=info:doi/10.1007/BF02795342&rft.aulast=Ahlfors&rft.aufirst=Lars V.&rfr_id=info:sid/en.wikipedia.org:Weil–Petersson metric" class="Z3988">
- 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 0084.28102413-419&rft.pub=Secrétariat Mathématique&rft.date=1958&rft_id=https://zbmath.org/?format=complete&q=an:0084.28102#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0124485#id-name=MR&rft.aulast=Weil&rft.aufirst=André&rfr_id=info:sid/en.wikipedia.org:Weil–Petersson metric" class="Z3988">
- 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 0537935381-389&rft.pub=Springer-Verlag&rft.date=1979&rft.isbn=978-0-387-90330-9&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=537935#id-name=MR&rft.aulast=Weil&rft.aufirst=André&rft_id=https://books.google.com/books?id=iYiiD9oKnBUC&rfr_id=info:sid/en.wikipedia.org:Weil–Petersson metric" class="Z3988">
- 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 249779147-64&rft.pub=Eur. Math. Soc., Zürich&rft.date=2009&rft_id=info:arxiv/0801.0175&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=2497791#id-name=MR&rft_id=info:doi/10.4171/055-1/2&rft.isbn=978-3-03719-055-5&rft.aulast=Wolpert&rft.aufirst=Scott A.&rfr_id=info:sid/en.wikipedia.org:Weil–Petersson metric" class="Z3988">
- 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