---
abstract: |
    We consider a Riemannian cylinder $\Omega$ endowed with a closed potential $1$-form $A$ and study the magnetic Laplacian $\Delta_A$ with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.

    *2000 Mathematics Subject Classification. 58J50, 35P15.*

    *Key words and phrases. Magnetic Laplacian, Eigenvalues, Upper and lower bounds, Zero magnetic field*
author:
- Bruno Colbois and Alessandro Savo
bibliography:
- 'CS.bib'
title: 'Lower bounds for the first eigenvalue of the magnetic Laplacian\'
---

Introduction {#intro}
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Let $(\Omega,g)$ be a compact Riemannian manifold with boundary. Consider the trivial complex line bundle $\Omega\times\bf C$ over $\Omega$; its space of sections can be identified with $C^{\infty}(\Omega,\bf C)$, the space of smooth complex valued functions on $\Omega$. Given a smooth real 1-form $A$ on $\Omega$ we define a connection $\nabla^A$ on $C^{\infty}(\Omega,\bf C)$ as follows: $$\label{connection}
\nabla^A_Xu=\nabla_Xu-iA(X)u$$ for all vector fields $X$ 