Surfaces with Commuting Boundary Laplacian and Dirichlet-to-Neumann Map

For $M\subset {\mathbb{R}}^{d\ge 3}$ a smooth, connected, compact $d$-dimensional submanifold with boundary, equipped with the standard metric, the Laplacian on $\mathrm{\partial }M$ is known to commute with the corresponding Dirichlet-to-Neumann map if and only if $M$ is a ball. In this paper, we investigate the $d=2$ case and show that, surprisingly, there exists a one-parameter family of submanifolds of ${\mathbb{R}}^2$ as above for which the boundary Laplacian and the Dirichlet-to-Neumann map commute, thus answering an open problem posed by Girouard, Karpukhin, Levitin, and Polterovich. We then classify all such Riemannian surfaces of genus $0$ or whose boundary has $k\ge 3$ connected components.