On the Friedlander–Nadirashvili invariants of surfaces

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 ∆ 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.

Degenerating sequences of conformal classes and the conformal Steklov spectrum

“…The following theorem establishes the correspondence between and the limit of when the sequence of conformal classes degenerates (see Section 4 for the definition). It is an analog of [KM20, Theorem 2.8] for the Steklov setting.…”

“…The first Friedlander–Nadirashvili invariant of a closed manifold was introduced in the paper [FN99] in 1999. The k th Nadirashvili–Friedlander invariant of a closed surface has been recently studied in the paper [KM20].…”

“…We proceed with the discussion of the functional I σ k . Unlike Theorem 1.4 in [KM20] Theorem 1.5 says nothing about conformal classes on which the value I σ k (Σ) is attained. We conjecture that Conjecture 1.6 The infimum I σ k (Σ) is attained if and only if Σ is diffeomorphic to the disc D 2 .…”

“…We proceed with the discussion of the functional . Unlike Theorem 1.4 in [KM20], Theorem 1.5 says nothing about conformal classes on which the value is attained. We conjecture that…”

Conformally maximal metrics for Laplace eigenvalues on surfaces

“…Consider a degenerating sequence C n of conformal classes of the genus 2 surface Σ 2 , converging (on the Deligne-Mumford compactification of the moduli space) to two copies of the equilateral torus. Using the continuity result [KM,Theorem 2.8] one has that…”