On the First Critical Field in Ginzburg-Landau Theory for Thin Shells and Manifolds

In this article, we investigate the response of a thin superconducting shell to an arbitrary external magnetic field. We identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H (c1) in Ginzburg-Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a simply connected surface of revolution and vertical and constant field, we further determine the exact number of vortices in the sample as the intensity of the applied field is raised just above H (c1). Finally, we derive via I"-convergence similar statements for three-dimensional domains of small thickness, where in this setting point vortices are replaced by vortex lines.