Existence of metrics maximizing the first eigenvalue on non-orientable surfaces

Henrik Matthiesen,Anna Siffert

doi:10.4171/jst/372

Abstract

We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in \cite{MS3}.

Highlights

For a closed Riemannian surface (Σ, g) the spectrum of the Laplace operator acting on smooth functions, is purely discrete and can be written as 0 = λ0 < λ1(Σ, g) ≤ λ2(Σ, g) ≤ λ3(Σ, g) ≤ · · · → ∞, where we repeat an eigenvalue as often as its multiplicity requires
The pioneering work of Hersch [Her70] and Yang–Yau [YY80] raised the natural question, whether there are metrics g that maximize the scale-invariant quantities λ1(Σ) := λ1(Σ, g) area(Σ, g) if Σ is a closed surface of fixed topological type
Such maximizers have remarkable properties. They always arise as immersed minimal surfaces in a sphere [ESI00] and are unique in their conformal class unless they are branched immersions into the two sphere [CKM17, MR96, NS19]

Summary

Introduction

The pioneering work of Hersch [Her70] and Yang–Yau [YY80] raised the natural question, whether there are metrics g that maximize the scale-invariant quantities λ1(Σ) := λ1(Σ, g) area(Σ, g) if Σ is a closed surface of fixed topological type (see [Kar[16], LY82] for the case of non-orientable surfaces). If Λ1(γ − 1) < Λ1(γ), there is a metric g on Σ = Σγ, which is smooth away from finitely many conical singularities, such that λ1(Σ, g) area(Σ, g) = Λ1(γ) In [MS19b], by means of a much more complicated glueing construction and using Theorem 1.1 and Theorem 1.2, we prove all of the spectral gap conditions assumed in Theorem 1.1 and Theorem 1.2 above In particular this implies the existence of maximizing metrics on closed surfaces of any topological type.

Compactness for non-orientable surfaces

Maximizing the first eigenvalue