Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds

TL;DR

This paper establishes monotonicity and sharp bounds for Steklov eigenvalues on space forms and warped product manifolds, advancing understanding of spectral geometry in curved spaces.

Contribution

It introduces new monotonicity results for normalized Steklov spectra and confirms a conjecture on eigenvalue bounds in warped product manifolds.

Findings

Monotonicity of normalized spectra for second and fourth-order Steklov problems on 2D geodesic disks.

Derivation of sharp eigenvalue bounds on warped product manifolds with non-negative Ricci curvature.

Confirmation of a conjecture on the lower bound of the first non-zero eigenvalue in 3D warped products.

Abstract

In the first part, we derive monotonicity of the normalized spectra for the second-order Steklov problem and two fourth-order Steklov problems on the $2$-dimensional geodesic disks with respect to the geodesic radius in the sphere and the hyperbolic space. The normalizations are made using four natural geometric factors. As corollaries, we get Escobar-type bounds for Steklov eigenvalues on $2$-dimensional geodesic disks with varying curvature in space forms. We also get two monotonicity results for higher-dimensional cases. In the second part, we obtain some sharp bounds concerning the spectra of the two fourth-order Steklov problems on warped product manifolds with non-negative Ricci curvature and a strictly convex boundary. In particular, we confirm Qiaoling Wang and Changyu Xia's conjecture (2018) on the sharp lower bound of the first non-zero eigenvalue of a fourth-order Steklov problem in the case of $3$-dimensional warped product manifolds.