On the Friedlander–Nadirashvili invariants of surfaces

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 Ik(M) indexed by positive integers k. In the present paper the values of these invariants on surfaces are investigated. We show that Ik(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 Ik(M) > Ik(S2). We also discuss the connection between the Friedlander–Nadirashvili invariants and the theory of cobordisms, and conjecture that Ik(M) is a cobordism invariant. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.