Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces

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].