Non-maximality of known extremal metrics on torus and Klein bottle

El Soufi-Ilias' theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of extremal metrics. We investigate the maximality of these metrics and prove that all of them are not maximal.Comment: Title changed to coincide with published version. Minor mistakes corrected and journal reference adde