Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

The question of determining for which eigenvalues there exists an eigenfunction has the same number nodal domains as label associated eigenvalue (Courant-sharp property) was motivated by analysis minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, M\"obius strips,\ldots . A natural toy model further investigations is flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly bottle square torus, whose higher multiplicities. this note, we prove that only Courant-sharp torus (resp. fundamental domain) are first second eigenvalues. We also consider cylinders $(0,\pi) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ radius circle $\mathbb{S}^1_r$, show Dirichlet these