Velling-Kirillov metric on the universal Teichmuller curve

Details Velling-Kirillov metric on the universal Teichmuller curve Velling-Kirillov metric on the universal Teichmuller curve

2002-06-19

arxiv.org/abs/math/0206202v3

Mathematics

Complex Variables

35 pages

Scientific paper

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+(S^1)/S^1 coincides with the Kirillov metric. We show that the vertical integration of the square of the symplectic form of 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.

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