标题： 具有交换边界Laplace算子和Dirichlet-to-Neumann映射的曲面 显示英文标题

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

摘要： 对于一个光滑、连通、紧致的带边流形$M\subset \mathbb{R}^{d\geq 3}$，配备标准度量，在$d$-维情况下，$\partial M$上的拉普拉斯算子与相应的狄利克雷到诺依曼映射可交换当且仅当$M$是球体。本文研究了$d=2$情况，并证明令人惊讶的是，存在一个单参数族的上述$\mathbb{R}^2$的子流形，其中边界上的拉普拉斯算子和狄利克雷到诺依曼映射是可交换的，从而回答了由Girouard、Karpukhin、Levitin和Polterovich提出的一个公开问题。 我们接着将所有这类亏格为$0$或边界有$k\geq 3$个连通分支的黎曼曲面分类。 显示英文摘要

摘要： 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.