What is the Morse index of the Scherk surfaces?

Question

The Scherk surfaces are properly embedded, complete minimal surfaces \begin{equation} S_\alpha \subset \mathbf{R}^3 \end{equation} that are asymptotic at infinity to the union of two planes $\Pi_1, \Pi_2 \subset \mathbf{R}^3$ which form the angle $\alpha$ along their intersection.

It is relatively obvious that they have infinite Morse index. However, one can quotient $S_\alpha$ by the discrete group $\mathbf{Z}$ corresponding to its periodicity along the axis $\Pi_1 \cap \Pi_2$, yielding a minimal surface \begin{equation} \Sigma_\alpha := S_\alpha / \mathbf{Z} \subset \mathbf{R}^2 \times \mathbf{R} / \mathbf{Z}. \end{equation}

What is the Morse index of $\Sigma_\alpha$? Does it have Jacobi fields in addition to those generated by isometries of $\mathbf{R}^3$ fixing $\Pi_1 \cap \Pi_2$?

Answer 1

These questions are indeed answered by the paper of Montiel and Ros that Otis suggested in his comment. In fact you can ask more generally for the index and $L^\infty$ nullity of $S_\alpha / m\mathbf{Z}$ for any integer (number of fundamental periods) $m \geq 1$. Corollary 15 of the Montiel-Ros paper (which does not mention the Scherk surfaces specifically) implies that the index is is $2m-1$ and the nullity $3$, independently of the value of $\alpha$.