Curvatures on the Teichmüller curve

Details Curvatures on the Teichmüller curve Curvatures on the Teichmüller curve

2010-05-13

arxiv.org/abs/1005.2356v1

Mathematics

Differential Geometry

16 pages

Scientific paper

The Teichm\"{u}ller curve is the fiber space over Teichm\"{u}ller space of closed Riemann surfaces, where the fiber over a point in Teichm\"{u}ller space is the underlying surface. We derive formulas for sectional curvatures on the Teichm\"{u}ller curve. In particular, our method can be applied to investigate the geometry of the Weil-Petersson geodesic as a three-manifold, and the degeneration of the curvatures near the infinity of the augmented Teichm\"{u}ller space along a Weil-Petersson geodesic, as well as the minimality of hyperbolic surfaces in this three-manifold.

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