The Velling-Kirillov metric on the universal Teichmüller curve

Abstract

We extend Velling's approach and prove that the second variation of the spherical areas of a family of domains defines a Hermitian metric on the universal Teichmuller curve, whose pull-back to Diff +(S1)/S1 coincides with the Kirillov metric. We call this Hermitian metric the Velling-Kirillov metric. We show that the vertical integration of the square of the symplectic form of the Velling-Kirillov metric on the universal Teichmuller curve is the symplectic form that defines the Weil-Petersson metric on the universal Teichmuller space. Restricted to a finite dimensional Teichmuller space, the vertical integration of the corresponding form on the Teichmuller curve is also the symplectic form that defines the Weil-Petersson metric on the Teichmuller space.