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A. Weil introduced a [[Kähler metric|Kähler metric]] for the [[Teichmüller space|Teichmüller space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300601.png" />, the space of homotopy-marked Riemann surfaces (cf. [[Riemann surface|Riemann surface]]) of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300602.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300603.png" /> punctures and negative [[Euler characteristic|Euler characteristic]], [[#References|[a1]]]. The cotangent space at a marked Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300604.png" /> (the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300605.png" /> of holomorphic quadratic differentials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300606.png" />; cf. also [[Quadratic differential|Quadratic differential]]) is considered with the Petersson Hermitian pairing. The Weil–Petersson metric calibrates the variations of the complex structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300607.png" />. The uniformization theorem implies that for a surface of negative Euler characteristic, the following two determinations are equivalent: a complex structure and a complete hyperbolic metric. Accordingly, the Weil–Petersson metric has been studied through [[Quasi-conformal mapping|quasi-conformal mapping]], solution of the inhomogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w1300609.png" />-equation (cf. also [[Neumann d-bar problem|Neumann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006010.png" />-problem]]), the prescribed curvature equation, and global analysis, [[#References|[a1]]], [[#References|[a7]]], [[#References|[a10]]].
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The quotient of the Teichmüller space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006011.png" /> by the action of the mapping class group is the moduli space of Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006012.png" /> (cf. also [[Moduli of a Riemann surface|Moduli of a Riemann surface]]; [[Moduli theory|Moduli theory]]); the Weil–Petersson metric is a mapping class group invariant and descends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006013.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006014.png" /> (the stable-curve compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006015.png" />) is a projective variety with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006016.png" /> (the divisor of noded stable-curves, i.e. the Riemann surfaces  "with disjoint simple loops collapsed to points"  and each component of the nodal-complement having negative Euler characteristic). Expansions for the Weil–Petersson metric in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006017.png" /> provide that the metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006018.png" /> is not complete and that there is a distance completion separating points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006019.png" />, [[#References|[a6]]].
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The Weil–Petersson metric has negative sectional curvature, [[#References|[a9]]], [[#References|[a12]]]. The behaviour near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006020.png" /> shows that the sectional curvature has as infimum negative infinity and as supremum zero. The holomorphic sectional, Ricci and scalar curvatures are each bounded above by genus-dependent negative constants. A modification of the metric introduced by C.T. McMullen [[#References|[a5]]] is Kähler-hyperbolic in the sense of M. Gromov (cf. also [[Gromov hyperbolic space|Gromov hyperbolic space]]), has positive first eigenvalue and provides that the sign of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006021.png" /> orbifold Euler characteristic is given by the parity of the dimension.
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A. Weil introduced a [[Kähler metric|Kähler metric]] for the [[Teichmüller space|Teichmüller space]] $T _ { g , n }$, the space of homotopy-marked Riemann surfaces (cf. [[Riemann surface|Riemann surface]]) of genus $g$ with $n$ punctures and negative [[Euler characteristic|Euler characteristic]], [[#References|[a1]]]. The cotangent space at a marked Riemann surface $\{ R \}$ (the space $Q ( R )$ of holomorphic quadratic differentials on $R$; cf. also [[Quadratic differential|Quadratic differential]]) is considered with the Petersson Hermitian pairing. The Weil–Petersson metric calibrates the variations of the complex structure of $\{ R \}$. The uniformization theorem implies that for a surface of negative Euler characteristic, the following two determinations are equivalent: a complex structure and a complete hyperbolic metric. Accordingly, the Weil–Petersson metric has been studied through [[Quasi-conformal mapping|quasi-conformal mapping]], solution of the inhomogeneous $\overline { \partial }$-equation (cf. also [[Neumann d-bar problem|Neumann $\overline { \partial }$-problem]]), the prescribed curvature equation, and global analysis, [[#References|[a1]]], [[#References|[a7]]], [[#References|[a10]]].
  
The Weil–Petersson Kähler form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006022.png" /> appears in several contexts. L.A. Takhtayan and P.G. Zograf [[#References|[a8]]] considered the local index theorem for families of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006023.png" />-operators and calculated the first Chern form of the determinant line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006024.png" /> using Quillen's construction of a metric based on the hyperbolic metric; the Chern form is
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The quotient of the Teichmüller space $\mathcal{T}_{g,n} $ by the action of the mapping class group is the moduli space of Riemann surfaces ${\cal M} _ { g , n }$ (cf. also [[Moduli of a Riemann surface|Moduli of a Riemann surface]]; [[Moduli theory|Moduli theory]]); the Weil–Petersson metric is a mapping class group invariant and descends to ${\cal M} _ { g , n }$. $\overline { \mathcal{M}_ { g , n } }$ (the stable-curve compactification of ${\cal M} _ { g , n }$) is a projective variety with $\mathcal{D} _ { g , n } = \overline { \mathcal{M} _ { g , n } } - \mathcal{M} _ { g , n }$ (the divisor of noded stable-curves, i.e. the Riemann surfaces  "with disjoint simple loops collapsed to points" and each component of the nodal-complement having negative Euler characteristic). Expansions for the Weil–Petersson metric in a neighbourhood of $\mathcal{D} _ { g , n }$ provide that the metric on ${\cal M} _ { g , n }$ is not complete and that there is a distance completion separating points on $\overline { \mathcal{M}_ { g , n } }$, [[#References|[a6]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006025.png" /></td> </tr></table>
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The Weil–Petersson metric has negative sectional curvature, [[#References|[a9]]], [[#References|[a12]]]. The behaviour near $\mathcal{D} _ { g , n }$ shows that the sectional curvature has as infimum negative infinity and as supremum zero. The holomorphic sectional, Ricci and scalar curvatures are each bounded above by genus-dependent negative constants. A modification of the metric introduced by C.T. McMullen [[#References|[a5]]] is Kähler-hyperbolic in the sense of M. Gromov (cf. also [[Gromov hyperbolic space|Gromov hyperbolic space]]), has positive first eigenvalue and provides that the sign of the ${\cal M} _ { g , n }$ orbifold Euler characteristic is given by the parity of the dimension.
  
The "universal curve"  is the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006026.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006027.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006028.png" /> above the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006029.png" />. The uniformization theorem provides a metric for the vertical line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006030.png" /> of the fibration. The setup extends to the compactification: The pushdown of the square of the first Chern form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006031.png" /> for the hyperbolic metric is the current class of
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The Weil–Petersson Kähler form $\omega _{\text{WP}}$ appears in several contexts. L.A. Takhtayan and P.G. Zograf [[#References|[a8]]] considered the local index theorem for families of $\overline { \partial }$-operators and calculated the first Chern form of the determinant line bundle $\operatorname{det} \; \operatorname{ind} \overline { \partial }$ using Quillen's construction of a metric based on the hyperbolic metric; the Chern form is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006032.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { 12 \pi ^ { 2 } } \omega _{\text{WP}}. \end{equation*}
  
[[#References|[a14]]]. This result is the basis for a proof of the projectivity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006033.png" />, [[#References|[a16]]].
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The  "universal curve"  is the fibration $\mathcal{C} _ { g ,\, n }$ over $\mathcal{T}_{g,n} $ with fibre $R$ above the class $\{ R \}$. The uniformization theorem provides a metric for the vertical line bundle ${\cal V} _ { g , n }$ of the fibration. The setup extends to the compactification: The pushdown of the square of the first Chern form of $\overline { \mathcal{V}_{ g , n} }$ for the hyperbolic metric is the current class of
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\begin{equation*} \frac { 1 } { 2 \pi ^ { 2 } } \omega_{ \text{WP}}, \end{equation*}
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[[#References|[a14]]]. This result is the basis for a proof of the projectivity of $\overline { \mathcal{M}_ { g , n } }$, [[#References|[a16]]].
  
 
The Weil–Petersson volume element appears in the calculation by E. D'Hoker and D.H. Phong [[#References|[a3]]] of the partition function integrand for the string theory of A.M. Polyakov. Generating functions have also been developed for the volumes of moduli spaces, [[#References|[a4]]], [[#References|[a17]]].
 
The Weil–Petersson volume element appears in the calculation by E. D'Hoker and D.H. Phong [[#References|[a3]]] of the partition function integrand for the string theory of A.M. Polyakov. Generating functions have also been developed for the volumes of moduli spaces, [[#References|[a4]]], [[#References|[a17]]].
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J.F. Brock considers a coarse combinatorial estimate for the Weil–Petersson distance in terms of the edge path metric in the pants complex, [[#References|[a18]]].
 
J.F. Brock considers a coarse combinatorial estimate for the Weil–Petersson distance in terms of the edge path metric in the pants complex, [[#References|[a18]]].
  
W. Fenchel and J. Nielsen presented  "twist-length"  coordinates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006034.png" />, as the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006035.png" /> for assembling pairs of pants, three-holed spheres with hyperbolic metric and geodesic boundaries, to form hyperbolic surfaces. The Kähler form has a simple expression in terms of these coordinates:
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W. Fenchel and J. Nielsen presented  "twist-length"  coordinates for $\mathcal{T}_{g,n} $, as the parameters $\{ ( \tau _ { j } , \text{l} _ { j } ) \}$ for assembling pairs of pants, three-holed spheres with hyperbolic metric and geodesic boundaries, to form hyperbolic surfaces. The Kähler form has a simple expression in terms of these coordinates:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006036.png" /></td> </tr></table>
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\begin{equation*} \omega _ { \text{WP} } = \Sigma _ { j } d \text{l} _ { j } \bigwedge d \tau _ { j }, \end{equation*}
  
[[#References|[a13]]]. Each geodesic length function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006037.png" /> is convex along Weil–Petersson geodesics, [[#References|[a15]]]. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006038.png" /> has an exhaustion by compact Weil–Petersson convex sets, [[#References|[a15]]].
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[[#References|[a13]]]. Each geodesic length function $\mathbf{l}_{*}$ is convex along Weil–Petersson geodesics, [[#References|[a15]]]. Consequently, $\mathcal{T}_{g,n} $ has an exhaustion by compact Weil–Petersson convex sets, [[#References|[a15]]].
  
A. Verjovsky and S. Nag [[#References|[a11]]] considered the Weil–Petersson geometry for the infinite-dimensional universal Teichmüller space and found that the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006039.png" /> coincides with the Kirillov–Kostant symplectic structure coming from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006040.png" />. I. Biswas and Nag [[#References|[a2]]] showed that the analogue of the Takhtayan–Zograf result above is valid for the universal moduli space obtained from the inductive limit of Teichmüller spaces for characteristic coverings.
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A. Verjovsky and S. Nag [[#References|[a11]]] considered the Weil–Petersson geometry for the infinite-dimensional universal Teichmüller space and found that the form $\omega _{\text{WP}}$ coincides with the Kirillov–Kostant symplectic structure coming from $\operatorname {Diff}^ { + } ( \mathbf{S} ^ { 1 } ) / \operatorname { Mob } ( \mathbf{S} ^ { 1 } )$. I. Biswas and Nag [[#References|[a2]]] showed that the analogue of the Takhtayan–Zograf result above is valid for the universal moduli space obtained from the inductive limit of Teichmüller spaces for characteristic coverings.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Some remarks on Teichmüller's space of Riemann surfaces"  ''Ann. of Math.'' , '''74''' :  2  (1961)  pp. 171–191</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Biswas,  S. Nag,  "Weil–Petersson geometry and determinant bundles on inductive limits of moduli spaces" , ''Lipa's legacy (New York, 1995)'' , Amer. Math. Soc.  (1997)  pp. 51–80</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. D'Hoker,  D.H. Phong,  "Multiloop amplitudes for the bosonic Polyakov string"  ''Nucl. Phys. B'' , '''269''' :  1  (1986)  pp. 205–234</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Kaufmann,  Yu. Manin,  D. Zagier,  "Higher Weil–Petersson volumes of moduli spaces of stable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006041.png" />-pointed curves"  ''Commun. Math. Phys.'' , '''181''' :  3  (1996)  pp. 763–787</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C.T. McMullen,  "The moduli space of Riemann surfaces is Kähler hyperbolic"  ''Ann. of Math.'' , '''151''' :  1  (2000)  pp. 327–357</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Masur,  "Extension of the Weil–Petersson metric to the boundary of Teichmuller space"  ''Duke Math. J.'' , '''43''' :  3  (1976)  pp. 623–635</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Nag,  "The complex analytic theory of Teichmüller spaces" , Wiley/Interscience  (1988)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.A. Takhtajan,  P.G. Zograf,  "A local index theorem for families of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006042.png" />-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces"  ''Commun. Math. Phys.'' , '''137''' :  2  (1991)  pp. 399–426</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.J. Tromba,  "On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Petersson metric"  ''Manuscripta Math.'' , '''56''' :  4  (1986)  pp. 475–497</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.J. Tromba,  "Teichmüller theory in Riemannian geometry" , Birkhäuser  (1992)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  S. Nag,  A. Verjovsky,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130060/w13006043.png" /> and the Teichmüller spaces"  ''Commun. Math. Phys.'' , '''130''' :  1  (1990)  pp. 123–138</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S.A. Wolpert,  "Chern forms and the Riemann tensor for the moduli space of curves"  ''Invent. Math.'' , '''85''' :  1  (1986)  pp. 119–145</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  S.A. Wolpert,  "On the Weil–Petersson geometry of the moduli space of curves"  ''Amer. J. Math.'' , '''107''' :  4  (1985)  pp. 969–997</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  S.A. Wolpert,  "The hyperbolic metric and the geometry of the universal curve"  ''J. Differential Geom.'' , '''31''' :  2  (1990)  pp. 417–472</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  S.A. Wolpert,  "Geodesic length functions and the Nielsen problem"  ''J. Differential Geom.'' , '''25''' :  2  (1987)  pp. 275–296</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  S.A. Wolpert,  "On obtaining a positive line bundle from the Weil–Petersson class"  ''Amer. J. Math.'' , '''107''' :  6  (1985)  pp. 1485–1507</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  P. Zograf,  "The Weil–Petersson volume of the moduli space of punctured spheres" , ''Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen/Seattle, WA, 1991)'' , Amer. Math. Soc.  (1993)  pp. 367–372</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  J.F. Brock,  "The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores"  ''Preprint''  (2001)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  L.V. Ahlfors,  "Some remarks on Teichmüller's space of Riemann surfaces"  ''Ann. of Math.'' , '''74''' :  2  (1961)  pp. 171–191</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Biswas,  S. Nag,  "Weil–Petersson geometry and determinant bundles on inductive limits of moduli spaces" , ''Lipa's legacy (New York, 1995)'' , Amer. Math. Soc.  (1997)  pp. 51–80</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E. D'Hoker,  D.H. Phong,  "Multiloop amplitudes for the bosonic Polyakov string"  ''Nucl. Phys. B'' , '''269''' :  1  (1986)  pp. 205–234</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R. Kaufmann,  Yu. Manin,  D. Zagier,  "Higher Weil–Petersson volumes of moduli spaces of stable $n$-pointed curves"  ''Commun. Math. Phys.'' , '''181''' :  3  (1996)  pp. 763–787</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C.T. McMullen,  "The moduli space of Riemann surfaces is Kähler hyperbolic"  ''Ann. of Math.'' , '''151''' :  1  (2000)  pp. 327–357</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  H. Masur,  "Extension of the Weil–Petersson metric to the boundary of Teichmuller space"  ''Duke Math. J.'' , '''43''' :  3  (1976)  pp. 623–635</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  S. Nag,  "The complex analytic theory of Teichmüller spaces" , Wiley/Interscience  (1988)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  L.A. Takhtajan,  P.G. Zograf,  "A local index theorem for families of $\overline { \partial }$-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces"  ''Commun. Math. Phys.'' , '''137''' :  2  (1991)  pp. 399–426</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A.J. Tromba,  "On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Petersson metric"  ''Manuscripta Math.'' , '''56''' :  4  (1986)  pp. 475–497</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  A.J. Tromba,  "Teichmüller theory in Riemannian geometry" , Birkhäuser  (1992)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S. Nag,  A. Verjovsky,  "$\operatorname{Diff}( S ^ { 1 } )$ and the Teichmüller spaces"  ''Commun. Math. Phys.'' , '''130''' :  1  (1990)  pp. 123–138</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  S.A. Wolpert,  "Chern forms and the Riemann tensor for the moduli space of curves"  ''Invent. Math.'' , '''85''' :  1  (1986)  pp. 119–145</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  S.A. Wolpert,  "On the Weil–Petersson geometry of the moduli space of curves"  ''Amer. J. Math.'' , '''107''' :  4  (1985)  pp. 969–997</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  S.A. Wolpert,  "The hyperbolic metric and the geometry of the universal curve"  ''J. Differential Geom.'' , '''31''' :  2  (1990)  pp. 417–472</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  S.A. Wolpert,  "Geodesic length functions and the Nielsen problem"  ''J. Differential Geom.'' , '''25''' :  2  (1987)  pp. 275–296</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  S.A. Wolpert,  "On obtaining a positive line bundle from the Weil–Petersson class"  ''Amer. J. Math.'' , '''107''' :  6  (1985)  pp. 1485–1507</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  P. Zograf,  "The Weil–Petersson volume of the moduli space of punctured spheres" , ''Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen/Seattle, WA, 1991)'' , Amer. Math. Soc.  (1993)  pp. 367–372</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  J.F. Brock,  "The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores"  ''Preprint''  (2001)</td></tr></table>

Latest revision as of 19:05, 16 June 2024

A. Weil introduced a Kähler metric for the Teichmüller space $T _ { g , n }$, the space of homotopy-marked Riemann surfaces (cf. Riemann surface) of genus $g$ with $n$ punctures and negative Euler characteristic, [a1]. The cotangent space at a marked Riemann surface $\{ R \}$ (the space $Q ( R )$ of holomorphic quadratic differentials on $R$; cf. also Quadratic differential) is considered with the Petersson Hermitian pairing. The Weil–Petersson metric calibrates the variations of the complex structure of $\{ R \}$. The uniformization theorem implies that for a surface of negative Euler characteristic, the following two determinations are equivalent: a complex structure and a complete hyperbolic metric. Accordingly, the Weil–Petersson metric has been studied through quasi-conformal mapping, solution of the inhomogeneous $\overline { \partial }$-equation (cf. also Neumann $\overline { \partial }$-problem), the prescribed curvature equation, and global analysis, [a1], [a7], [a10].

The quotient of the Teichmüller space $\mathcal{T}_{g,n} $ by the action of the mapping class group is the moduli space of Riemann surfaces ${\cal M} _ { g , n }$ (cf. also Moduli of a Riemann surface; Moduli theory); the Weil–Petersson metric is a mapping class group invariant and descends to ${\cal M} _ { g , n }$. $\overline { \mathcal{M}_ { g , n } }$ (the stable-curve compactification of ${\cal M} _ { g , n }$) is a projective variety with $\mathcal{D} _ { g , n } = \overline { \mathcal{M} _ { g , n } } - \mathcal{M} _ { g , n }$ (the divisor of noded stable-curves, i.e. the Riemann surfaces "with disjoint simple loops collapsed to points" and each component of the nodal-complement having negative Euler characteristic). Expansions for the Weil–Petersson metric in a neighbourhood of $\mathcal{D} _ { g , n }$ provide that the metric on ${\cal M} _ { g , n }$ is not complete and that there is a distance completion separating points on $\overline { \mathcal{M}_ { g , n } }$, [a6].

The Weil–Petersson metric has negative sectional curvature, [a9], [a12]. The behaviour near $\mathcal{D} _ { g , n }$ shows that the sectional curvature has as infimum negative infinity and as supremum zero. The holomorphic sectional, Ricci and scalar curvatures are each bounded above by genus-dependent negative constants. A modification of the metric introduced by C.T. McMullen [a5] is Kähler-hyperbolic in the sense of M. Gromov (cf. also Gromov hyperbolic space), has positive first eigenvalue and provides that the sign of the ${\cal M} _ { g , n }$ orbifold Euler characteristic is given by the parity of the dimension.

The Weil–Petersson Kähler form $\omega _{\text{WP}}$ appears in several contexts. L.A. Takhtayan and P.G. Zograf [a8] considered the local index theorem for families of $\overline { \partial }$-operators and calculated the first Chern form of the determinant line bundle $\operatorname{det} \; \operatorname{ind} \overline { \partial }$ using Quillen's construction of a metric based on the hyperbolic metric; the Chern form is

\begin{equation*} \frac { 1 } { 12 \pi ^ { 2 } } \omega _{\text{WP}}. \end{equation*}

The "universal curve" is the fibration $\mathcal{C} _ { g ,\, n }$ over $\mathcal{T}_{g,n} $ with fibre $R$ above the class $\{ R \}$. The uniformization theorem provides a metric for the vertical line bundle ${\cal V} _ { g , n }$ of the fibration. The setup extends to the compactification: The pushdown of the square of the first Chern form of $\overline { \mathcal{V}_{ g , n} }$ for the hyperbolic metric is the current class of

\begin{equation*} \frac { 1 } { 2 \pi ^ { 2 } } \omega_{ \text{WP}}, \end{equation*}

[a14]. This result is the basis for a proof of the projectivity of $\overline { \mathcal{M}_ { g , n } }$, [a16].

The Weil–Petersson volume element appears in the calculation by E. D'Hoker and D.H. Phong [a3] of the partition function integrand for the string theory of A.M. Polyakov. Generating functions have also been developed for the volumes of moduli spaces, [a4], [a17].

J.F. Brock considers a coarse combinatorial estimate for the Weil–Petersson distance in terms of the edge path metric in the pants complex, [a18].

W. Fenchel and J. Nielsen presented "twist-length" coordinates for $\mathcal{T}_{g,n} $, as the parameters $\{ ( \tau _ { j } , \text{l} _ { j } ) \}$ for assembling pairs of pants, three-holed spheres with hyperbolic metric and geodesic boundaries, to form hyperbolic surfaces. The Kähler form has a simple expression in terms of these coordinates:

\begin{equation*} \omega _ { \text{WP} } = \Sigma _ { j } d \text{l} _ { j } \bigwedge d \tau _ { j }, \end{equation*}

[a13]. Each geodesic length function $\mathbf{l}_{*}$ is convex along Weil–Petersson geodesics, [a15]. Consequently, $\mathcal{T}_{g,n} $ has an exhaustion by compact Weil–Petersson convex sets, [a15].

A. Verjovsky and S. Nag [a11] considered the Weil–Petersson geometry for the infinite-dimensional universal Teichmüller space and found that the form $\omega _{\text{WP}}$ coincides with the Kirillov–Kostant symplectic structure coming from $\operatorname {Diff}^ { + } ( \mathbf{S} ^ { 1 } ) / \operatorname { Mob } ( \mathbf{S} ^ { 1 } )$. I. Biswas and Nag [a2] showed that the analogue of the Takhtayan–Zograf result above is valid for the universal moduli space obtained from the inductive limit of Teichmüller spaces for characteristic coverings.

References

[a1] L.V. Ahlfors, "Some remarks on Teichmüller's space of Riemann surfaces" Ann. of Math. , 74 : 2 (1961) pp. 171–191
[a2] I. Biswas, S. Nag, "Weil–Petersson geometry and determinant bundles on inductive limits of moduli spaces" , Lipa's legacy (New York, 1995) , Amer. Math. Soc. (1997) pp. 51–80
[a3] E. D'Hoker, D.H. Phong, "Multiloop amplitudes for the bosonic Polyakov string" Nucl. Phys. B , 269 : 1 (1986) pp. 205–234
[a4] R. Kaufmann, Yu. Manin, D. Zagier, "Higher Weil–Petersson volumes of moduli spaces of stable $n$-pointed curves" Commun. Math. Phys. , 181 : 3 (1996) pp. 763–787
[a5] C.T. McMullen, "The moduli space of Riemann surfaces is Kähler hyperbolic" Ann. of Math. , 151 : 1 (2000) pp. 327–357
[a6] H. Masur, "Extension of the Weil–Petersson metric to the boundary of Teichmuller space" Duke Math. J. , 43 : 3 (1976) pp. 623–635
[a7] S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley/Interscience (1988)
[a8] L.A. Takhtajan, P.G. Zograf, "A local index theorem for families of $\overline { \partial }$-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces" Commun. Math. Phys. , 137 : 2 (1991) pp. 399–426
[a9] A.J. Tromba, "On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Petersson metric" Manuscripta Math. , 56 : 4 (1986) pp. 475–497
[a10] A.J. Tromba, "Teichmüller theory in Riemannian geometry" , Birkhäuser (1992)
[a11] S. Nag, A. Verjovsky, "$\operatorname{Diff}( S ^ { 1 } )$ and the Teichmüller spaces" Commun. Math. Phys. , 130 : 1 (1990) pp. 123–138
[a12] S.A. Wolpert, "Chern forms and the Riemann tensor for the moduli space of curves" Invent. Math. , 85 : 1 (1986) pp. 119–145
[a13] S.A. Wolpert, "On the Weil–Petersson geometry of the moduli space of curves" Amer. J. Math. , 107 : 4 (1985) pp. 969–997
[a14] S.A. Wolpert, "The hyperbolic metric and the geometry of the universal curve" J. Differential Geom. , 31 : 2 (1990) pp. 417–472
[a15] S.A. Wolpert, "Geodesic length functions and the Nielsen problem" J. Differential Geom. , 25 : 2 (1987) pp. 275–296
[a16] S.A. Wolpert, "On obtaining a positive line bundle from the Weil–Petersson class" Amer. J. Math. , 107 : 6 (1985) pp. 1485–1507
[a17] P. Zograf, "The Weil–Petersson volume of the moduli space of punctured spheres" , Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen/Seattle, WA, 1991) , Amer. Math. Soc. (1993) pp. 367–372
[a18] J.F. Brock, "The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores" Preprint (2001)
How to Cite This Entry:
Weil-Petersson metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil-Petersson_metric&oldid=23133
This article was adapted from an original article by Scott A. Wolpert (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article