2016/11/30 by Xiaolong Han, Han, Xiaolong
Mathematics · #35J05 #35P15 #58J50 #Advanced Algebra and Geometry #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1611.09959
openalex publication_date 2016/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on the manifold is analytic. We prove that if the eigenfunctions are equidistributed at a small scale, then the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold. In particular, on the negatively curved manifolds with analytic metric and on the tori, we show that in any eigenbasis, there is a full density subsequence of eigenfunctions such that the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold.