Eigenvalue bounds of the Robin Laplacian with magnetic field

On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues ( λ k ( τ , α ) ) k of the magnetic Laplacian with Robin boundary conditions. Here, τ is a positive number that defines the Robin condition and α is a real differential 1-form on M that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter τ , and a lower bound of the Ricci curvature of M (see Theorem 1.3 and Corollary 1.5). The main technique is to use the Bochner formula established in Egidi et al. (Ricci curvature and eigenvalue estimates for the magentic Laplacian on manifolds, arXiv:1608.01955v1 ) for the magnetic Laplacian and to integrate it over M (see Theorem 1.2). In the last part, we compare the eigenvalues λ k ( τ , α ) with the first eigenvalue λ 1 ( τ ) = λ 1 ( τ , 0 ) (i.e. without magnetic field) and the Neumann eigenvalues λ k ( 0 , α ) (see Theorem 1.6) using the min-max principle.