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

Abstract

In this paper, we establish a quantitative lower bound, expressed in terms of the isoperimetric deficit, for the normalized first eigenvalue deficit of the Robin Laplacian with a negative boundary parameter on bounded horospherically convex domains in hyperbolic space. As a consequence, the geodesic ball maximizes the first Robin eigenvalue within this class, thereby resolving the higher-dimensional counterpart of an open problem posed by Celentano, Krejčiřík and Lotoreichik in \cite{CKL26}. Furthermore, for a positive boundary parameter, we derive a quantitative upper bound for the normalized difference between the first Robin eigenvalue of such a domain and that of the geodesic ball with the same perimeter.