In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space

Teichmüller space

In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...

 T_g,n of genus g Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

s with n marked points. It was introduced by using the Petersson inner product on forms on a Riemann surface (introduced by Hans Petersson

Hans Petersson

Hans Petersson was a German mathematician. He introduced the Petersson inner product and is also known for the Ramanujan–Petersson conjecture.-References:...

).

Definition

If a point of Teichmüller space is represented by a Riemann surface R, then the cotangent space at that point can be identified with the space of quadratic differential

Quadratic differential

In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle.If the section is holomorphic, then the quadratic differentialis said to be holomorphic...

s at R. Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric.

Properties

stated, and proved, that the Weil–Petersson metric is a Kähler metric. proved that it has negative holomorphic sectional, scalar, and Ricci curvatures. The Weil–Petersson metric is usually not complete.

Generalizations

The Weil–Petersson metric can be defined in a similar was for some moduli spaces of higher-dimensional varieties.