A research brief on the Weil-Petersson metric

Scott A. Wolpert

March 15, 2000

A. Weil introduced a Kähler metric for the Teichmüller space T_{g,n}, the space of homotopy marked Riemann surfaces of genus g with n punctures and negative Euler characteristic,[1]. The cotangent space at a marked Riemann surface {R} the space Q(R) of holomorphic quadratic differentials on R is considered with the Petersson hermitian pairing. The Weil-Petersson metric calibrates the variations of the complex structure of {R}. For a surface of negative Euler characteristic by the Uniformization Theorem two determinations are equivalent: a complex structure and a complete hyperbolic metric. Accordingly the Weil-Petersson metric has been studied through quasiconformal maps, solutions of the inhomogeneous ¯∂-equation, the prescribed curvature equation and global analysis, [1, 7, 11].

The quotient of the Teichmüller space T_{g,n} by the action of the mapping class group is the moduli space of Riemann surfaces M_{g,n}; the Weil-Petersson metric is mapping class group invariant and descends to M_{g,n}. \overline{M_{g,n}} the stable-curve compactification of M_{g,n} is a projective variety with D_{g,n} = \overline{M_{g,n}} − 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 neighborhood of D_{g,n} provide that the metric on M_{g,n} is not complete and that there is a distance completion separating points on \overline{M_{g,n}}, [5].

The Weil-Petersson metric has negative sectional curvature, [10, 14]. The behavior near D_{g,n} provides that the sectional curvature has infimum negative infinity and 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 is Kählerhyperbolic in the sense of M. Gromov, has positive first eigenvalue, and provides that the sign of the M_{g,n} orbifold Euler characteristic is given by the parity of the dimension, [6].

The Weil-Petersson Kähler form ω_{W P} appears in several contexts. L. A. Takhtajan and P. G. Zograf considered the local index theorem for families of ¯∂-operators and calculated the first Chern form of the determinant line bundle det ind ¯∂ with D. G. Quillen’s construction of a metric based on the hyperbolic metric; the Chern form is \frac{1}{12π^{2}} ω_{W P}, [9]. The “universal curve” is the fibration C_{g,n} over T_{g,n} with fibre R above the class {R}. The Uniformization Theorem provides a metric for the vertical line bundle V_{g,n} of the fibration. The setup extends to the compactification: the pushdown of the square of the first Chern form of \overline{V_{g,n}} for the hyperbolic metric is the current class of \frac{1}{2π^{2}} ω_{W P},[16]. The result is the basis for a proof of the projectivity of \overline{M_{g,n}},[12]. The Weil-Petersson volume element appears in the calculation by E. D’Hoker and D. H. Phong of the partition function integrand for A. M. Polyakov’s String Theory, [3]. Generating functions have also been developed for the volumes of moduli spaces, [4, 17].

Fenchel-Nielsen presented “twist-length” coordinates for T_{g,n} as the parameters {(τ_{j}, 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 the coordinates ω_{W P} = Σ_{j} dℓ_{j} ∧ dτ_{j}, [13]. Each geodesic length function ℓ_{∗} is convex along Weil-Petersson geodesics, [15]. In consequence T_{g,n} has an exhaustion by compact Weil-Petersson convex sets, [15].

A. Verjovsky and S. Nag considered the Weil-Petersson geometry for the infinite dimensional universal Teichmüller space and found that the form ω_{W P} coincides with the Kirillov-Kostant symplectic structure coming from Diff^{+}(𝕊^{1})/Mob(𝕊^{1}), [8]. I. Biswas and S. Nag showed that the analog of the Takhtajan-Zograf result is valid for the universal moduli space obtained from the inductive limit of Teichmüller spaces for characteristic coverings, [2].

References

[1] Lars V. Ahlfors. Some remarks on Teichmüller’s space of Riemann surfaces. Ann. of Math. (2), 74:171–191, 1961.

[2] Indranil Biswas and Subhashis Nag. Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces. In Lipa’s legacy (New York, 1995), pages 51–80. Amer. Math. Soc., Providence, RI, 1997.

[3] Eric D’Hoker and D. H. Phong. Multiloop amplitudes for the bosonic Polyakov string. Nuclear Phys. B, 269(1):205–234, 1986.

[4] R. Kaufmann, Yu. Manin, and D. Zagier. Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Comm. Math. Phys., 181(3):763–787, 1996.

[5] Howard Masur. Extension of the Weil-Petersson metric to the boundary of Teichmuller space. Duke Math. J., 43(3):623–635, 1976.

[6] Curtis T. McMullen. The moduli space of Riemann surfaces is Kähler hyperbolic. preprint, 1998.

[7] Subhashis Nag. The complex analytic theory of Teichmüller spaces. John Wiley & Sons Inc., New York, 1988. A Wiley-Interscience Publication.

[8] Subhashis Nag and Alberto Verjovsky. Diff(S^{1}) and the Teichmüller spaces. Comm. Math. Phys., 130(1):123–138, 1990.

[9] L. A. Takhtajan and P. G. Zograf. A local index theorem for families of \overline{∂}-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces. Comm. Math. Phys., 137(2):399–426, 1991.

[10] 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):475–497, 1986.

[11] Anthony J. Tromba. Teichmüller theory in Riemannian geometry. Birkhäuser Verlag, Basel, 1992. Lecture notes prepared by Jochen Denzler.

[12] Scott A. Wolpert. On obtaining a positive line bundle from the Weil-Petersson class. Amer. J. Math., 107(6):1485–1507 (1986), 1985.

[13] Scott A. Wolpert. On the Weil-Petersson geometry of the moduli space of curves. Amer. J. Math., 107(4):969–997, 1985.

[14] Scott A. Wolpert. Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math., 85(1):119–145, 1986.

[15] Scott A. Wolpert. Geodesic length functions and the Nielsen problem. J. Differential Geom., 25(2):275–296, 1987.

[16] Scott A. Wolpert. The hyperbolic metric and the geometry of the universal curve. J. Differential Geom., 31(2):417–472, 1990.

[17] Peter Zograf. The Weil-Petersson volume of the moduli space of punctured spheres. In Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), pages 367–372. Amer. Math. Soc., Providence, RI, 1993.