Ahlfors Bers Colloquium
May
19-22, 2005
University of
Michigan
Ann Arbor, Michigan
Registration
Organizers
Dick
Canary
Juha
Heinonen
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Workshop
Abstract Detail
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David Dumas Rice University
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Grafting, the schwarzian derivative, and complex projective structures on surfaces
We present results comparing the complex-analytic and hyperbolic-geometric aspects of complex projective structures on Riemann surfaces. The complex-analytic perspective is based on the schwarzian derivative, whereby the complex projective structures on a Riemann surface correspond to holomorphic quadratic differentials. The geometric perspective is based on a fundamental result of Thurston which states that every complex projective structure is obtained in a unique way by grafting a hyperbolic surface along a measured geodesic lamination. Our main result is that the schwarzian derivative and the grafting lamination of a projective structure are approximately related by the familiar construction in which the trajectories of a quadratic differential are straightened with respect to a hyperbolic metric to yield a measured geodesic lamination. Specifically, we show that the holomorphic quadratic differential whose trajectories represent the grafting lamination and the schwarzian derivative of a projective structure have bounded difference. We also discuss applications of this result to the holonomy of projective structures, and its connection to our previous results on the asymptotics behavior of Thurston's grafting coordinates and the antipodal involution on measured laminations. | ||
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