Surfaces with Commuting Boundary Laplacian and Dirichlet-to-Neumann Map
Romain Speciel
TL;DR
The study characterizes when the boundary Laplacian commutes with the Dirichlet-to-Neumann map for compact 2D manifolds with boundary. By exploiting the conformal covariance of the Laplacian in dimension two and reducing to boundary Fourier data via a Riemann mapping, the authors show that commutativity enforces a finite Fourier support on the reciprocal conformal factor, yielding a complete classification for genus zero or multiple boundary components: $M$ is $\sigma$-isometric to a disc, a logarithmic oval, or a flat cylinder. A striking outcome is the existence of a one-parameter family of planar domains—the logarithmic ovals—that satisfy the commuting property, and the results sharpen GKLP22 by removing the boundary-connectedness assumption and clarifying the doubly connected case. An open problem remains for manifolds with genus $g>0$ and exactly two boundary components. All mathematical notation is kept within $...$ delimiters.
Abstract
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.