Teichmüller curves in genus three and just likely intersections in Gmn×Gan

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmuller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmuller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmuller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).