A lower bound for the nodal sets of Steklov eigenfunctions

TL;DR

This paper establishes a lower bound on the size of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds, showing that their measure grows at least as a specific power of the eigenvalue.

Contribution

It provides the first explicit lower bound for the measure of nodal sets of Steklov eigenfunctions depending only on the eigenvalue and the manifold.

Findings

Lower bound of nodal set measure: $H^{n-1}(N_)=\geq C\lambda^{\frac{3-n}{2}}$

Bound depends only on the eigenvalue and the manifold

Applicable when zero is a regular value of the eigenfunctions

Abstract

We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_\lambda$ be its nodal set. Assume that zero is a regular value of Steklov eigenfunctions. We show that $$H^{n-1}(N_\lambda)\geq C\lambda^{\frac{3-n}{2}}$$ for some positive constant $C$ depending only on the manifold.