Linearized Ginzburg-Landau equation on a curved surface and curvature-induced superconductivity enhancement

We derive the linearized Ginzburg-Landau (GL) equation on a spatially curved surface in the presence of a magnetic field. By introducing a novel transverse order parameter with extremely slow varying along the surface, we decouple the linearized GL equation into a surface part and a transverse part, and obtain the superconducting geometric potential (GP) under the superconducting/vacuum boundary condition. Further, we numerically study the nucleation of a rectangle thin superconducting film bent around the surface of a cylinder. Our results show that, the geometric curvature can weaken the effect of the magnetic field, and the competition between the applied magnetic field and the superconducting GP will lead to an abnormal $ T_c(B) $ dependence in the GP dominant region and a $ T_c $ enhancement when the magnetic field dominates.