Upper bounds on eigenvalue multiplicities for surfaces of genus $0$ revisited (Work in progress)

Work in progress. We will gratefully welcome any comment or suggestion, in particular towards a detailed proof of the inequality $\mathrm{mult}(\lambda_k) \leq (2k-3)$ for planar domainsWe revisit two 1999 papers: [1]~ \emph{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} ~~and~~ [2]~ \emph{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 surface $M$ is bounded from above by $(2k-3)$ provided that $k \geq 3$. Here, $M$ is [1] a closed surface with genus $0$, or [2] a planar 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. In the case of closed surfaces with genus $0$, we provide full details of the proof that $\mathrm{mult}(\lambda_k) \leq (2k-3)$ by introducing and carefully studying the \emph{combinatorial type} (defined in Subsection~5.2) of the nodal sets involved. In the case of planar domains, we consider the three boundary conditions, Dirichlet, Neumann, Robin, and we also carefully study the \emph{combinatorial types} of the nodal sets involved. We provide full details of the proof of the inequality $\mathrm{mult}(\lambda_k) \leq (2k-2)$ ($k \geq 3$). Unfortunately, we are so far unable to complete the proof of the inequality $\mathrm{mult}(\lambda_k) \leq (2k-3)$, even when the domain is simply connected. In the last section, we explain the difficulties we met. Our work on the subject is still ongoing