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

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 ${\lambda }_1(M)$ 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).