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

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, Krejčiřík 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.