On the Friedlander–Nadirashvili invariants of surfaces

Abstract

Let M be a closed smooth manifold. In 1999, Friedlander and Nadirashvili introduced a new differential invariant I₁ (M) using the first normalized nonzero eigenvalue of the Lalpace–Beltrami operator Δ_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 Δ_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) = Ik(S²) unless M is a non-orientable surface of even genus. For orientable surfaces and k = 1 this was earlier shown by Petrides. In fact Friedlander and Nadirashvili suggested that I₁(M) = I₁ (S²) for any surface M different from RP². 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(S²). 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.

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