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 Δ𝑔 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 Δ𝑔 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.