Surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map
Romain Speciel · Journal of Spectral Theory · 2026
For M\subset \mathbb{R}^{d\geq 3} a smooth, connected, compact d -dimensional submanifold with boundary, equipped with the standard metric, the Laplacian on \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\geq 3 connected components.