# Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

## TL;DR

This paper establishes sharp bounds for the first Robin Laplacian eigenvalue on Riemannian manifolds, extending Euclidean results and analyzing geometric influences, with applications to hyperbolic space and space forms.

## Contribution

It provides new sharp bounds for Robin eigenvalues on Riemannian manifolds, extends monotonicity results to revolution manifolds, and proves uniform lower bounds in hyperbolic space.

## Key findings

- Sharp lower bounds for positive Robin parameters depending on geometry.
- Extension of eigenvalue monotonicity to revolution manifolds.
- Uniform lower bound for hyperbolic space domains with large boundary parameter.

## Abstract

We consider the first eigenvalue $\lambda_1(\Omega,\sigma)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with smooth boundary, $\sigma\in\bf R$ being the Robin boundary parameter. When $\sigma>0$ we give a positive, sharp lower bound of $\lambda_1(\Omega,\sigma)$ in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of $\Omega$, a lower bound of the mean curvature of $\partial\Omega$ and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates. Then, we extend a monotonicity result for $\lambda_1(\Omega,\sigma)$ obtained in Euclidean space by Giorgi and Smits to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that $\lambda_1(\Omega,\sigma)$ is uniformly bounded below by $\frac{(n-1)^2}4$ for all bounded domains in the hyperbolic space of dimension $n$, provided that the boundary parameter $\sigma\geq\frac{n-1}{2}$ (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed