Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds

TL;DR

This paper establishes geometric bounds for the Steklov eigenvalues on differential forms in warped product manifolds, extending known results for functions and exploring topological influences.

Contribution

It provides new eigenvalue bounds for differential forms on warped product manifolds, including Escobar type bounds and bounds for hypersurfaces of revolution.

Findings

Derived lower bounds for Steklov eigenvalues in warped products with non-negative Ricci curvature.

Obtained sharp bounds for hypersurfaces of revolution.

Highlighted the impact of topology on the spectrum of differential forms.

Abstract

We consider the Steklov problem on differential $p$-forms defined by M. Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., $0$-forms), highlighting the influence of the underlying topology on the spectrum for $p$-forms in general.