Eigenvalue bounds of the Robin Laplacian with magnetic field

On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(\lambda\_k(\tau,\alpha))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $\tau$ is a positive number that defines the Robin condition and $\alpha$ 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 $\tau$ and a lower bound of the Ricci curvature of $M$ (see Theorem \ref{estimate1} and Corollary \ref{corestimate}). The main technique is to use the Bochner formula established in \cite{ELMP} for the magnetic Laplacian and to integrate it over $M$ (see Theorem \ref{bochnermagnetic1}). In the last part, we compare the eigenvalues $\lambda\_k(\tau,\alpha)$ with the first eigenvalue $\lambda\_1(\tau)=\lambda\_1(\tau,0)$ (i.e. without magnetic field) and the Neumann eigenvalues $\lambda\_k(0,\alpha)$ (see Theorem \ref{thm:comp}) using the min-max principle.