Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces

TL;DR

This paper proves the existence of metrics that maximize the first Laplace eigenvalue normalized by area on all closed surfaces, resolving a long-standing problem and providing new proofs and techniques for both orientable and nonorientable cases.

Contribution

The authors establish the existence of maximizing metrics for the first Laplace eigenvalue on all closed surfaces, including nonorientable ones, and introduce refined methods for proving strict monotonicity under topological modifications.

Findings

Existence of maximizing metrics on all closed surfaces.

Strict monotonicity of the normalized eigenvalue under topological changes.

New simplified proof for the orientable case.

Abstract

Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof of existence in the orientable case complementing that of [Pet24b], thus resolving the long-standing existence problem for $\lambda_1$-maximizing metrics on closed surfaces of any topology. Namely, we prove by contradiction that the supremum $\Lambda_1(M)$ of the normalized first eigenvalue over all metrics on $M$ obeys the strict monotonicity $\Lambda_1(M\#\mathbb{RP}^2)>\Lambda_1(M)$ and $\Lambda_1(M\#\mathbb{T}^2)>\Lambda_1(M)$ under the attachment of cross-caps and handles, via a substantial refinement of techniques introduced in [KKMS24].