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We revisit two papers which appeared in 1999: [1] M. Hoffmann-Ostenhof, T.Hoffmann-Ostenhof, and N. Nadirashvili. On the multiplicity of eigenvalues ofthe Laplacian on surfaces. Ann. Global Anal. Geom. 17 (1999) 43--48. [2] T.Hoffmann-Ostenhof, P. Michor, and N. Nadirashvili. Bounds on the multiplicityof eigenvalues for fixed membranes. Geom. Funct. Anal. 9 (1999) 1169--1188. Themain 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], $M$ ishomeomorphic to a sphere. In [2], $M$ is a plane domain with Dirichlet boundarycondition. In both cases, the starting label of eigenvalues is $1$. The proofsgiven in [1,2] are not very detailed, and often rely on figures or specialconfigurations of nodal sets. The purpose of this monograph is to providedetailed general proofs for the above upper bounds and to extend the results toRobin boundary conditions. When $M$ is homeomorphic to a sphere, we provide acomplete proof that $\mathrm{mult}(\lambda_k) \le (2k-3)$ for any $k\ge 3$, byintroducing and carefully studying the combinatorial type and a labeling of thenodal domains of some particular eigenfunctions. When $M$ is a plane domain, weconsider Dirichlet and Robin boundary conditions and we also study thecombinatorial types and a labeling of the nodal domains of some particulareigenfunctions. We prove the inequality $\mathrm{mult}(\lambda_k) \le (2k-2)$for general $C^{\infty}$ bounded domains and all $k \ge 3$. We prove theinequality $\mathrm{mult}(\lambda_k) \le (2k-3)$ for $k \ge 3$ under theadditional assumption that the domain is simply connected.

Upper bounds on eigenvalue multiplicities for spheres and plane domains revisited