Arbitrarily Many Non-degenerate Local Maxima of First Nonzero Eigenfunctions on Positively Curved Two-spheres

Abstract

We prove that, for every integer $m\ge 2$, there is a smooth closed Riemannian surface $(M,g)$ diffeomorphic to $\mathbb S^2$, with positive Gaussian curvature, such that $λ_1(M)$ is simple and, after choosing the sign, its normalized first nonzero Laplace eigenfunction has at least $m$ distinct non-degenerate local maxima. This gives a negative answer to the Open Question of Grossi and Provenzano (Math. Ann. 389(4): 3447--3470, 2024).