  It was proved by Montiel and Ros that for each conformal structure on a
compact surface there is at most one metric which admits a minimal immersion
into some unit sphere by first eigenfunctions. We generalize this theorem to
the setting of metrics with conical singularities induced from branched minimal
immersions by first eigenfunctions into spheres. Our primary motivation is the
fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are
induced by minimal branched immersions into spheres. In particular, we show
that the properties of such metrics induced from $\mathbb{S}^2$ differ
significantly from the properties of those induced from $\mathbb{S}^m$ with
$m>2$. This feature appears to be novel and needs to be taken into account in
the existing proofs of the sharp upper bounds for the first non-zero eigenvalue
of the Laplacian on the $2$-torus and the Klein bottle. In the present paper we
address this issue and give a detailed overview of the complete proofs of these
upper bounds following the works of Nadirashvili,
Jakobson-Nadirashvili-Polterovich, El Soufi-Giacomini-Jazar, Nadirashvili-Sire
and Petrides.
