Upper bounds on eigenvalue multiplicities for spheres and plane domains revisited

Abstract: We revisit two papers which appeared in 1999: [1] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili. On the multiplicity of eigenvalues of the Laplacian on surfaces. Ann. Global Anal. Geom. 17 (1999) 43--48. [2] T. Hoffmann-Ostenhof, P. Michor, and N. Nadirashvili. Bounds on the multiplicity of eigenvalues for fixed membranes. Geom. Funct. Anal. 9 (1999) 1169--1188. The main result of these papers is that the multiplicity $\mathrm{mult}(\lambda_k(M))$ of the $k$th eigenvalue of the Riemannian surface $M$ is bounded from above by $(2k-3)$ provided that $k \ge 3$. In [1], $M$ is homeomorphic to a sphere. In [2], $M$ is a plane domain with Dirichlet boundary condition. In both cases, the starting label of eigenvalues is $1$. The proofs given in [1,2] are not very detailed, and often rely on figures or special configurations of nodal sets. The purpose of this monograph is to provide detailed general proofs for the above upper bounds and to extend the results to Robin boundary conditions. When $M$ is homeomorphic to a sphere, we provide a complete proof that $\mathrm{mult}(\lambda_k) \le (2k-3)$ for any $k\ge 3$, by introducing and carefully studying the combinatorial type and a labeling of the nodal domains of some particular eigenfunctions. When $M$ is a plane domain, we consider Dirichlet and Robin boundary conditions and we also study the combinatorial types and a labeling of the nodal domains of some particular eigenfunctions. We prove the inequality $\mathrm{mult}(\lambda_k) \le (2k-2)$ for general $C<sup>{\infty}$ bounded domains and all $k \ge 3$. We prove the inequality $\mathrm{mult}(\lambda_k) \le (2k-3)$ for $k \ge 3$ under the additional assumption that the domain is simply connected.