The second Robin eigenvalue in non-compact rank-1 symmetric spaces

TL;DR

This paper establishes a quantitative inequality for the second Robin eigenvalue in non-compact rank-1 symmetric spaces, showing geodesic balls maximize this eigenvalue among equal-volume domains under certain conditions.

Contribution

It generalizes previous Euclidean and hyperbolic results to a broader class of symmetric spaces, providing new spectral inequalities for the second Robin eigenvalue.

Findings

Geodesic balls maximize the second Robin eigenvalue among equal-volume domains.

The result applies to negative Robin parameters in a specific spectral regime.

Extension of Euclidean and hyperbolic eigenvalue inequalities to symmetric spaces.

Abstract

In this paper, we prove a quantitative spectral inequality for the second Robin eigenvalue in non-compact rank-1 symmetric spaces. In particular, this shows that for bounded domains in non-compact rank-1 symmetric spaces, the geodesic ball maximises the second Robin eigenvalue among domains of the same volume, with negative Robin parameter in the regime connecting the first nontrivial Neumann and Steklov eigenvalues. This result generalises the work of Freitas and Laugesen in the Euclidean setting [FL21] as well as our previous work in the hyperbolic space [LWW20].