Weil–Petersson metric

Weil–Petersson metric
Name	Weil–Petersson metric
Type	Kähler metric
Defined on	Teichmüller space, moduli space
Introduced by	André Weil, Robert Petersson
Applications	Teichmüller theory, hyperbolic geometry, string theory

Weil–Petersson metric. The Weil–Petersson metric is a Kähler metric on the Teichmüller space of Riemann surfaces that descends to the moduli space of curves. It arose in the work of André Weil and Robert Petersson and has been central to interactions between Riemann surface, Teichmüller theory, complex analysis, and algebraic geometry. Its study connects prominent figures and institutions such as William Thurston, Shing-Tung Yau, Curt McMullen, and research centers like the Institute for Advanced Study and IHÉS.

Definition and construction

The Weil–Petersson metric is defined on the tangent space of Teichmüller space via an L^2 pairing of harmonic Beltrami differentials with holomorphic quadratic differentials arising from the Hodge theory of compact Riemann surfaces. This construction uses techniques developed by Jean-Pierre Serre, W. V. D. Hodge, and later refinements by Kunihiko Kodaira and Donald C. Spencer. On a marked surface, one pairs cotangent vectors represented by quadratic differentials using the Petersson inner product originally considered in the work of Hans Petersson on automorphic forms and extended in the context of moduli problems by David Mumford. The metric is invariant under the action of the mapping class group studied by Max Dehn and Jakob Nielsen and descends to the Deligne–Mumford compactification introduced by Pierre Deligne and David Mumford.

Basic properties

The Weil–Petersson metric is Kähler but incomplete, with symmetries related to the mapping class group and properties influenced by the hyperbolic metrics of surfaces studied by Henri Poincaré and Carl Friedrich Gauss. It interacts with Hodge theory from Phillip Griffiths and has modulational behavior reflected in the work of Alexandre Grothendieck on stacks and by Pierre Deligne on monodromy. The metric admits a Weil–Petersson symplectic form connected to results of William Goldman and is sensitive to Fenchel–Nielsen coordinates developed by Werner Fenchel and Jakob Nielsen. Notable contributors to foundational properties include H. L. Royden, Scott A. Wolpert, and Anthony J. Tromba.

Relation to Teichmüller theory and moduli spaces

In Teichmüller theory the Weil–Petersson metric complements the Teichmüller metric studied by Oswald Teichmüller and is compared with Finsler structures considered by Lars Ahlfors and Lipman Bers. The metric encodes complex-analytic and algebraic-geometric data relevant to the moduli space of curves explored by David Mumford, Joe Harris, and by the techniques of Mikhail Gromov in geometric topology. Connections to the Deligne–Mumford compactification inform intersections with the work of Maxim Kontsevich and Edward Witten on intersection theory and enumerative geometry. The mapping class group dynamics addressed by Nikolai V. Ivanov and Howard Masur shape the global geometry relevant to moduli.

Curvature and geometric behavior

Curvature computations by Scott A. Wolpert and Anthony J. Tromba show that the Weil–Petersson metric has negative sectional curvature in many directions, with pinching and degeneration phenomena analyzed through methods of William Thurston and Maryam Mirzakhani. Asymptotic behavior near the boundary of moduli involves plumbing constructions used by Howard Masur and compactification techniques of Pierre Deligne and David Mumford. Results by Xiuxiong Chen and Simon Donaldson link curvature properties to stability in complex geometry, while comparisons to metrics studied by Shing-Tung Yau provide bridges to Calabi–Yau theory and moduli of higher-dimensional varieties.

Geodesics, completion, and metric completion

Studies of Weil–Petersson geodesics by Scott A. Wolpert, Howard Masur, and Jeffrey F. Brock reveal complex behavior including nonuniqueness and recurrence related to mapping class group pseudo-Anosov dynamics introduced by William Thurston. The metric completion corresponds to the augmented Teichmüller space and connects to the Deligne–Mumford compactification analyzed by William Abikoff and John H. Hubbard. Geodesic flow properties have been linked to ergodic theory by Alex Eskin and Maryam Mirzakhani and to coarse geometry perspectives developed by Howard Masur and Yair Minsky.

Applications and connections to other areas

The Weil–Petersson metric appears in quantum field theory contexts referenced by Edward Witten and in string theory moduli considerations examined by Joseph Polchinski and Nathan Seiberg. It plays a role in the study of three-manifolds via hyperbolic geometry related to William Thurston and in geometric group theory through mapping class group actions investigated by Benson Farb and Dan Margalit. Intersections with arithmetic geometry involve techniques from Gerd Faltings and Pierre Deligne, while computational aspects have been influenced by work at Mathematical Sciences Research Institute and by algorithms inspired by Robert C. Penner.

Category:Metrics on moduli spaces