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Schiffer variations and Abelian differentials

Thursday 30th July 2015 - 11:30 to 12:30
INI Seminar Room 1
Given a Riemann surface and an Abelian differential, we consider Cech style deformations based at zeros of the differential. Deformations are given in terms of slit mappings, degenerate Schwarz Christoffel mappings. We describe the associated deformation cocycles valued in vector fields.

Schiffer gave an exact formula for the change in the double pole Green's function corresponding to his conformal gluing deformation. We follow his approach and develop the second order variation formula for the double pole Green's function. Consequences are second order variation formulas for Abelian differentials and for the Riemann period matrix. The second variation of the period matrix is in the style of Rauch's celebrated formula and is given in terms of the 2-jet of the corresponding differentials at the base point zero.

Applications may include the Teichmuller geodesic flow on the space of Abelian differentials and the curvature of the Siegel upper half space metric on the image of Teichmuller space by the period matrix mapping.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons