Finiteness of algebraic Weil-Petersson totally geodesic curves in pointed moduli spaces

For every pair $(g,n)$ with $2g-2+n>0$, we prove that $\mathcal M_{g,n}$ contains only finitely many closed algebraic curves that are totally geodesic for the Weil-Petersson metric. We show that every such curve is an arithmetic quadratic Teichmüller curve and has equal heights for all marked maximal cylinders in every periodic direction. Although there are infinitely many arithmetic Teichmüller curves, we can bound the degree of the associated torus covering by the crucial equal-height property. This leads to the desired finiteness statement. We also prove that quadratic Teichmüller curves are characterized by the existence of a holomorphic splitting of the logarithmic cotangent Kodaira-Spencer map. This result is the analogue of Möller and Viehweg's characterization of Kobayashi geodesics in $\mathcal A_g$.