次黎曼流形上的粗节点计数
Coarse nodal counts on sub-Riemannian manifolds
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中文总结 AI 辅助
该研究针对次黎曼流形，证明了次拉普拉斯本征函数线性组合的节点集的粗Courant定理和Bézout定理，推测结果可推广到一般闭等正则次黎曼流形，方法结合拓扑持续性与Hörmander向量场的各向异性索伯列夫理论。
中文摘要 AI 辅助
我们研究次拉普拉斯算子本征函数线性组合的节点集的粗拓扑。更确切地说，我们证明了分层群商得到的紧致幂零流形上次拉普拉斯算子本征函数线性组合的Courant定理和Bézout定理的粗版本。我们推测这些结果可推广到一般闭等正则次黎曼流形，并概述了证明该推广的方案。所使用的方法结合了拓扑持续性与Hörmander向量场的各向异性索伯列夫理论，推广了最近在黎曼情形中已实现的思路。
英文摘要
We study coarse topology of nodal sets of linear combinations of eigenfunctions of sub-Laplacians. More precisely, we prove coarse versions of Courant's and Bézout's theorems for linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds obtained as quotients of stratified groups. We conjecture the extensions of these results to general closed equiregular sub-Riemannian manifolds and outline a programme for proving them. The method we use combines topological persistence and anisotropic Sobolev theory of Hörmander vector fields, generalizing the ideas which have recently been implemented in the Riemannian case.
