Review of Yau's conjecture on zero sets of Laplace eigenfunctions
Analysis of PDEs
2019-08-06 v1 Classical Analysis and ODEs
Complex Variables
Differential Geometry
Spectral Theory
Abstract
This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.
Keywords
Cite
@article{arxiv.1908.01639,
title = {Review of Yau's conjecture on zero sets of Laplace eigenfunctions},
author = {Alexander Logunov and Eugenia Malinnikova},
journal= {arXiv preprint arXiv:1908.01639},
year = {2019}
}