关于Laplace-Beltrami算子的Arnold横截性猜想
微分几何
2026-05-26 v4 偏微分方程分析
谱理论
摘要
本文关注连通流形上黎曼度量集合的结构，使得相应的Laplace-Beltrami算子具有给定重数的特征值。我们研究的出发点是Colin de Verdière引入的“强Arnold假设”，该假设认为更高重数的Laplace特征值在扰动下的分裂方式与对称矩阵的特征值完全相同。Colin de Verdière和Besson给出了满足强Arnold假设的度量的简单几何刻画。利用Besson的刻画，我们证明除了一个无限余维的集合外，所有度量都满足强Arnold假设，并由此得到具有任意给定重数特征值的度量集合的精确余维数。此外，我们证明对于所有具有至多六重特征值的度量，强Arnold假设成立，并讨论了几个违反该假设的度量例子。
引用
@article{arxiv.2312.16939,
title = {On Arnold's Transversality Conjecture for the Laplace--Beltrami Operator},
author = {Josef Greilhuber and Willi Kepplinger},
journal= {arXiv preprint arXiv:2312.16939},
year = {2026}
}
备注
41 pages, changes in v4: It came to our attention that the simplified transversality condition (Theorem 1.1) was previously obtained by G. Besson. We updated the preprint accordingly. Secondly, a discussion of the (non)-manifold structure of the set of metrics with an eigenvalue of a given multiplicity near metrics where transversality fails was added