Nodal sets of eigenfunctions of sub-Laplacians

TL;DR

This paper explores the properties of nodal sets of eigenfunctions of sub-Laplacians on compact manifolds, revealing anisotropic behaviors and providing bounds that challenge existing conjectures for classical Laplacians.

Contribution

It introduces the study of nodal sets for hypoelliptic operators, applying sub-Riemannian geometry tools and demonstrating that Yau's conjectured bounds do not hold for sub-Laplacians.

Findings

Nodal sets exhibit anisotropic behavior in sub-Riemannian settings.

Hausdorff measure bounds for nodal sets differ from classical Laplacian cases.

Yau's conjecture does not extend to sub-Laplacians on compact manifolds.

Abstract

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.