Current Developments in Mathematics

Volume 2018

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Pages: 179 – 212



Alexander Logunov (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Eugenia Malinnikova (Institute for Advanced Study, Princeton, New Jersey, U.S.A.; and Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway)


This is a review of old and new results and methods related to the Yau conjecture on the zero sets 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.

This work was completed during the time A.L. served as a Clay Research Fellow.

E.M. is supported by Project 275113 of the Research Council of Norway and NSF grant no. DMS-1638352.

Published 17 December 2019