Weil-Petersson metric

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)