Sharp estimates for the Robin Laplacian under a perimeter constraint in hyperbolic space

TL;DR

This paper derives sharp bounds for the first eigenvalue of the Robin Laplacian on horospherically convex domains in hyperbolic space, showing geodesic balls optimize these bounds and addressing an open problem.

Contribution

It provides the first isoperimetric-type bounds for Robin Laplacian eigenvalues in hyperbolic space, including both lower and upper bounds depending on the boundary parameter.

Findings

Geodesic balls maximize the first eigenvalue for negative boundary parameters.

Established lower bounds in terms of isoperimetric deficit.

Derived upper bounds for positive boundary parameters.

Abstract

In this paper, we establish a lower bound, in terms of the isoperimetric deficit, for the first eigenvalue of the Robin Laplacian with negative boundary parameter on horospherically convex bounded domains in the hyperbolic space. This implies that the geodesic ball maximizes this eigenvalue among all such domains, thereby providing a partial resolution to an open problem posed by Celentano, Krej\v{c}i\v{r}\'{i}k and Lotoreichik in \cite{CKL26}. Furthermore, we derive upper bounds for the first eigenvalue of the Robin Laplacian with positive boundary parameter on horospherically convex bounded domains in the hyperbolic space.