Minimal immersions by first eigenfunctions of Riemann surfaces with large conformal automorphism groups. The Klein quartic

Minimal immersions of closed surfaces into spheres by first eigenfunctions play an important role in both the theory of minimal surfaces and spectral geometry. In this paper, we establish the existence of equivariant conformal minimal immersions of this type for Hurwitz surfaces and other families of %Riemann surfaces with large automorphism group. Finally, we explicitly construct a conformal minimal equivariant immersion of the Klein quartic into the unit sphere $S^7$ by first eigenfunctions. Ths immersion is a natural candidate for maximizing the normalized ${\lambda }_1$ among closed genus-3 surfaces.