标题： 节点几何与低能本征函数的拓扑结构 显示英文标题

标题： Nodal geometry and topology of low energy eigenfunctions

摘要： 我们研究紧致黎曼流形和有界欧几里得域上拉普拉斯特征函数的节点几何和拓扑的各种方面，特别关注低频区域。 这包括节点集的几何和拓扑，特别是在佩恩性质、开角估计、(基本)谱间隙等区域内的表现，以及上述所有内容在小尺度扰动下的行为。 我们的目标是突出与基态/低能特征函数相关的谱理论和节点现象有趣方面，而不是渐近结果。 显示英文摘要

摘要： We investigate various aspects of the nodal geometry and topology of Laplace eigenfunctions on compact Riemannian manifolds and bounded Euclidean domains, with particular emphasis on the low frequency regime. This includes geometry and topology of nodal sets, particularly in and around the area of the Payne property, opening angle estimates, (fundamental) spectral gaps etc., and behaviour of all of the above under small scale perturbations. We aim to highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions, as opposed to asymptotic results.