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 S 2 differ significantly from the properties of those induced from 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,

Let M be a closed smooth manifold. In 1999, L. Friedlander and N. Nadirashvili introduced a new differential invariant I 1 (M) using the first normalized nonzero eigenvalue of the Lalpace-Beltrami operator ∆ g of a Riemannian metric g. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use k-th eigenvalues of ∆ g to define the invariants I k (M) indexed by positive integers k. In the present paper the values of these invariants on surfaces are investigated. We show that I k (M) = I k (S 2) unless M is a nonorientable surface of even genus. For orientable surfaces and k = 1 this was earlier shown by R. Petrides. In fact L. Friedlander and N. Nadirashvili suggested that I 1 (M) = I 1 (S 2) for any surface M different from RP 2. We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has I k (M) > I k (S 2). We also discuss the connection between the Friedlander-Nadirashvili invariants and the theory of cobordisms, and conjecture that I k (M) is a cobordism invariant.

Let Σ be a compact surface with boundary. For a given conformal class c on Σ the functional σ * k (Σ, c) is defined as the supremum of the k−th normalized Steklov eigenvalue over all metrics in c. We consider the behaviour of this functional on the moduli space of conformal classes on Σ. A precise formula for the limit of σ * k (Σ, cn) when the sequence {cn} degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander-Nadirashvili invariants of closed manifolds defined as infc σ * k (Σ, c), where the infimum is taken over all conformal classes c on Σ. We show that these quantities are equal to 2πk for any surface with boundary. As an application of our techniques we obtain new estimates on the k−th normalized Steklov eigenvalue of a non-orientable surface in terms of its genus and the number of boundary components.

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