Isoperimetric control of the Steklov spectrum

TL;DR

This paper establishes bounds on Steklov eigenvalues of domains in various manifolds using isoperimetric ratios, revealing uniform bounds in Euclidean, hyperbolic, and spherical spaces, and relating them to boundary Laplace eigenvalues.

Contribution

It introduces isoperimetric control of Steklov spectrum on manifolds with non-negative Ricci curvature and relates Steklov eigenvalues to boundary Laplace eigenvalues.

Findings

Normalized Steklov eigenvalues are bounded by isoperimetric ratios.

Uniform bounds for Steklov eigenvalues in Euclidean, hyperbolic, and spherical spaces.

Relationship between Steklov eigenvalues and boundary Laplace-Beltrami eigenvalues.

Abstract

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its bounding hypersurface.