Weil–Petersson metric
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 {{harvs|txt|authorlink=André Weil|first=André |last=Weil|year=1958|year2=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
{{harvtxt|Weil|1958}} stated, and {{harvtxt|Ahlfors|1961}} proved, that the Weil–Petersson metric is a Kähler metric. {{harvtxt|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.
See also
References
- {{Citation | last1=Ahlfors | first1=Lars V. | title=Some remarks on Teichmüller's space of Riemann surfaces | jstor=1970309 | mr=0204641 | year=1961 | journal=Annals of Mathematics |series=Second Series | volume=74 | issue=1 | pages=171–191 | doi=10.2307/1970309| hdl=2027/mdp.39015095258003 | hdl-access=free }}
- {{Citation | last1=Ahlfors | first1=Lars V. | title=Curvature properties of Teichmüller's space | doi=10.1007/BF02795342 | doi-access=free | mr=0136730 | year=1961b | journal=Journal d'Analyse Mathématique | volume=9 | pages=161–176| hdl=2027/mdp.39015095248350 | s2cid=124921349 | hdl-access=free }}
- {{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152à 168; 2e éd.corrigée, Exposé 168 | publisher=Secrétariat Mathématique | location=Paris | language=French | mr=0124485 |zbl=0084.28102 | year=1958 | chapter=Modules des surfaces de Riemann | pages=413–419}}
- {{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Scientific works. Collected papers. Vol. II (1951--1964) | orig-year=1958 | url=https://books.google.com/books?id=iYiiD9oKnBUC | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90330-9 | mr=537935 | year=1979 | chapter=On the moduli of Riemann surfaces | pages=381–389}}
- {{eom|id=Weil–Petersson_metric|first=Scott A.|last= Wolpert}}
- {{Citation | last1=Wolpert | first1=Scott A. | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. II | arxiv=0801.0175 | publisher=Eur. Math. Soc., Zürich | series=IRMA Lect. Math. Theor. Phys. | doi=10.4171/055-1/2 | mr=2497791 | year=2009 | volume=13 | chapter=The Weil-Petersson metric geometry | pages=47–64| isbn=978-3-03719-055-5 }}
- {{Citation | last1=Wolpert | first1=Scott A. | author1-link=Scott A. Wolpert| title=Families of Riemann Surfaces and Weil-Petersson Geometry | url=http://www.ams.org/bookstore-getitem/item=cbms-113 | publisher=Amer. Math. Soc., Providence, Rhode Island | series=CBMS Reg. Conf. Series in Math. | isbn=978-0-8218-4986-6 | mr=2641916 | year=2010 | volume=113 | number=113 | doi=10.1090/cbms/113| arxiv=1202.4078 | s2cid=7880175 }}
External links
{{DEFAULTSORT:Weil-Petersson metric}}