Measure of nodal sets of analytic Steklov eigenfunctions

Abstract

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