# Review of Yau's conjecture on zero sets of Laplace eigenfunctions

## TL;DR

This paper reviews historical and recent advances on Yau's conjecture concerning the zero sets of Laplace eigenfunctions, highlighting key results, methods, and the transition from real-analytic to smooth settings.

## Contribution

It synthesizes existing results and introduces new developments related to Yau's conjecture, especially in the smooth setting, emphasizing main ideas over technical details.

## Key findings

- Donnelly and Fefferman solved the conjecture for real-analytic manifolds.
- New results extend understanding of zero sets in smooth manifolds.
- Two-dimensional methods provide additional insights into zero set structure.

## 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.