# On the Friedlander-Nadirashvili invariants of surfaces

## TL;DR

This paper investigates the Friedlander-Nadirashvili invariants of surfaces, revealing that for non-orientable surfaces of even genus, these invariants exceed those of the sphere, and explores their potential as cobordism invariants.

## Contribution

It extends the understanding of Friedlander-Nadirashvili invariants to all surfaces, showing differences between orientable and non-orientable cases, and proposes their relation to cobordism theory.

## Key findings

- For non-orientable surfaces of even genus, invariants exceed those of the sphere.
- The invariants are equal to the sphere's for orientable surfaces and certain non-orientable cases.
- Conjecture that these invariants are cobordism invariants.

## Abstract

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 $\Delta_g$ 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 $\Delta_g$ 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 $I_k(M)=I_k(\mathbb{S}^2)$ unless $M$ is a non-orientable 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 $I_1(M)=I_1(\mathbb{S}^2)$ for any surface $M$ different from $\mathbb{RP}^2$. We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has $I_k(M)>I_k(\mathbb{S}^2)$. 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.