Curvatures on the Teichm\"uller curve

TL;DR

This paper derives formulas for sectional curvatures on the Teichmüller curve, enabling analysis of its geometric properties, including Weil-Petersson geodesics, curvature degeneration, and minimality of hyperbolic surfaces.

Contribution

It provides new formulas for curvatures on the Teichmüller curve and applies them to study its geometric structure and degeneration phenomena.

Findings

Formulas for sectional curvatures derived

Analysis of Weil-Petersson geodesic geometry

Curvature degeneration near augmented Teichmüller space boundary

Abstract

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.