The Work of Logunov and Malinnikova in Relation to Yau's Conjecture (Fall 2019)


We will discuss the recent work of Logunov and Malinnikova, for which they were awarded a Clay research award. This work led to the proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture. Nadirashvili's conjecture roughly states that if \(u\) is a harmonic function in the unit ball in \(\mathbb{R}^n\) which is zero at the center of the ball, then the \((n-1)\)-dimensional Hausdorff measure of the zero set of \(u\) in the ball is bounded below by a purely dimensional constant. Yau's conjecture concerns the \((n-1)\)-dimensional Hausdorff measure of the zero set of Laplace eigenfunctions. In particular, Yau conjectured that the zero set of the eigenfunction associated to the eigenvalue \(\lambda\) should have \((n-1)\)-dimensional Hausdorff measure which is approximately \(\sqrt{\lambda}\).