Indices 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 and $G\colon M\to S^2$ is the holomorphic map corresponding to $g$. We consider the meromorphic function $w$ on the Riemann surface $M_a=\left\{(z,w)\in\widehat{\mathbb{C}}^2 \mid w^2=z(z-a)\left(z+\frac{1}{a}\right)\right\}(a\geqslant 1 )$ homeomorphic to a torus. We find $a_0>1$ and determine the index of $tw$ for all $a$ in the range $1\leqslant a\leqslant a_0$ and all $t>0$.