the Weil-Petersson metric
/ VYE PEH-ter-son /
Teichmüller space is the space of hyperbolic shapes a surface can take, and like any space of shapes it begs for a notion of 'how far apart are two shapes' and 'which directions of deformation are big or small.' The Weil-Petersson metric is one natural answer: a Riemannian metric on Teichmüller space built by measuring deformations against the surface's own hyperbolic geometry. It is the metric that makes moduli space behave like a finite-volume negatively curved object and underlies much of the analysis of moduli of curves.
Precisely, a tangent vector to T(S) at a point X is an infinitesimal deformation of the complex structure, represented by a harmonic Beltrami differential mu (or dually by a holomorphic quadratic differential q, a section of the square of the cotangent bundle of X). The Weil-Petersson inner product of two such quadratic differentials q_1, q_2 is the integral over X of q_1 times the conjugate of q_2, divided by the hyperbolic area form: integral over X of (q_1 conj(q_2)) / (rho^2), where rho^2 is the hyperbolic metric. This pairing is Hermitian and defines a Kähler metric on T(S) — invariant under the mapping class group, hence descending to moduli space M(S).
Its geometry is rich and consequential: the Weil-Petersson metric has NEGATIVE sectional curvature (not bounded away from zero, and not constant), is geodesically convex, but is INCOMPLETE — geodesics can run off the edge in finite length, exactly where surfaces degenerate by pinching a geodesic to zero length. Wolpert showed the WP Kähler form equals (1/2) times the sum of d(l_i) ^ d(tau_i) in Fenchel-Nielsen coordinates, so lengths and twists are symplectically conjugate. The metric's finite total volume on moduli space (computed by Wolpert and Mirzakhani via beautiful recursion) underlies intersection-number formulas on the moduli of curves, linking hyperbolic geometry to algebraic geometry and even to 2D gravity.
On the genus-2 moduli space, head along a Weil-Petersson geodesic toward the boundary where one of the three pants curves shrinks: l_1 -> 0. The WP distance to that degenerate surface is FINITE (the metric is incomplete), even though the hyperbolic surface itself becomes a noded surface (a pinched curve) infinitely far in the Teichmüller metric. This finiteness is what gives M(S) finite Weil-Petersson volume, the quantity Mirzakhani computed by a recursion over pants decompositions.
WP incompleteness: pinching a curve (l_1 -> 0) sits at finite Weil-Petersson distance, giving moduli space finite WP volume.
The Weil-Petersson metric is incomplete and its curvature is negative but NOT bounded above by a negative constant nor below — do not treat it as a constant-curvature or even pinched-curvature space. It also differs from the Teichmüller metric (a Finsler, non-Riemannian metric that IS complete); the two are not the same and should not be confused.