Computation of the index of some meromorphic functions of degree 3 on tori

Abstract

The index of a meromorphic function $g$ on a compact Riemann surface is an invariant of $g$, which is defined as the number of negative eigenvalues of the differential operator $L:=-{\Delta}-|dG|^2$, where ${\Delta}$ is the Laplacian with respect to a conformal metric $ds^2$ on the Riemann surface, $G \colon M \to S^2$ is the holomorphic map corresponding to $g$. We consider the meromorphic function $w$ on the Riemann surface $M_a= \{(z,w) \in\widehat{\mathbb{C}}^2 \mid w^2=z(z-a)(z+\frac{1}{a})\}(a \geqslant 1 )$ homeomorphic to a torus, and we determine the index of $tw$ for all $a$ in the range $1 \leqslant a \leqslant a_0$ (where $a_0$ can be numerically evaluated) and all $t>0$.