Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds

TL;DR

This paper establishes bounds relating Steklov and Laplace eigenvalues on manifolds with boundary, under curvature conditions, and provides topology-based bounds in two dimensions without curvature assumptions.

Contribution

It introduces new bounds connecting Steklov and Laplace eigenvalues under curvature assumptions and offers topology-based bounds in 2D without curvature restrictions.

Findings

Steklov eigenvalues are controlled by boundary Laplace eigenvalues under positive curvature.

In 2D, Steklov eigenvalues are bounded by topological properties without curvature constraints.

Results extend understanding of spectral geometry on manifolds with boundary.

Abstract

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for these Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.