Nodal Sets of Eigenfunctions of Sub-Laplacians

被引:1
|
作者
Eswarathasan, Suresh [1 ]
Letrouit, Cyril [2 ,3 ]
机构
[1] Dalhousie Univ, Halifax, NS B3H 4R2, Canada
[2] MIT, Cambridge, MA 02139 USA
[3] Univ Paris Saclay, Lab Math Orsay, CNRS, UMR 8628, Batiment 307, F-91405 Orsay, France
基金
加拿大自然科学与工程研究理事会;
关键词
INEQUALITY;
D O I
10.1093/imrn/rnad219
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians. A standard example is the sum of squares of bracket-generating vector fields on compact quotients of the Heisenberg group. Our results show that nodal sets behave in an anisotropic way, which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation, and desingularization at singular points. Furthermore, we provide a simple example demonstrating that for sub-Laplacians, the Hausdorff measure of nodal sets of eigenfunctions cannot be bounded above by $\sqrt \lambda $, which is the bound conjectured by Yau for Laplace-Beltrami operators on smooth manifolds.
引用
收藏
页码:20670 / 20700
页数:31
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