Classification of Teichmüller curves
Final Report Abstract
Teichmüller curves are algebraic curves in the moduli spare of curves that are totally geodesic for the Teichmiiller metric. Interest in these curves stems from the search for billiard tables with optimal dynamical behavior. This project contributes to the classification and geometric understanding of Teichmiiller curves in two ways. First, the constraints to the existente of Teichmüller curves were improved from finiteness statements into a non-existence statements in specific low genus Gases. The combinatorial complexity is enourmous, the proof computer assisted, and raises doubts if all Teichmüller curves might ever be classified and listed completely. Second, this project contributes to geometric understanding of the first Teichmüller surface discovered by McMullen, Mukamel and Wright, the Gothic locus. In Joint vvork with Torres-Teigell we give a Hilbert modular form that cuts out the Teichmüller curves in this Gothic locus and thus enables to compute many of their characteristic invariants. This in turn enabled Torres-Teigell to compute a fundamental invariant, the Masur-Veech volume, of the Gothic locus.
Publications
- Euler characteristics of Gothic Teichmüller curves. Geom. Topol. 24 (2020), no. 3, 1149-1210
Martin Möller, David Torres-Teigell
(See online at https://doi.org/10.2140/gt.2020.24.1149) - Compactifying the rank two Hitchin system via spectral data on semistable curves
Johannes Horn, Martin Möller
(See online at https://doi.org/10.48550/arXiv.2201.08104)
