Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class

We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the $k$-th normalized Laplace-Beltrami eigenvalue functional $\bar\lambda_k$ in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain $\mathbb S^1$-equivariant harmonic maps $\mathbb T^2\to\mathbb S^3$. For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for $\bar\lambda_1$ in this conformal class with the value of $\bar\lambda_1$ being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for $\bar\lambda_1$ in its conformal class, then it is $\mathbb S^1$-equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in $\mathbb S^3$ called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals.