Measure of nodal sets of analytic Steklov eigenfunctions

TL;DR

This paper proves a conjecture relating the size of nodal sets of Steklov eigenfunctions on real analytic manifolds to their eigenvalues, establishing an upper bound proportional to the eigenvalue.

Contribution

It establishes an upper bound on the measure of nodal sets of Steklov eigenfunctions, confirming a conjecture by Lin and Bellova.

Findings

Nodal set measure is bounded by a constant times the eigenvalue.

The result applies to real analytic Riemannian manifolds with analytic boundary.

Confirms a conjecture in spectral geometry.

Abstract

Let $(\Omega, g)$ be a real analytic Riemannian manifold with real analytic boundary $\partial \Omega$. Let $\psi_{\lambda}$ be an eigenfunction of the Dirichlet-to-Neumann operator $\Lambda$ of $(\Omega, g, \partial \Omega)$ of eigenvalue $\lambda$. Let $\mathcal N_{\lambda_j}$ be its nodal set. Then $\mathcal H^{n-2} (\mathcal N_{\lambda}) \leq C_{g, \Omega} \lambda.$ This proves a conjecture of F. H. Lin and K. Bellova.