Minimal surfaces: A Lagrangian derivation of first and second variations

This article develops a rigorous Lagrangian formulation of variational calculus for minimal surfaces, using extensively the concept of pullback covariant derivative. The goal of this paper is not to present new results on this extensively studied problem, but rather to introduce geometrical arguments that simplify the calculations and eliminate the need for ad hoc assumptions throughout the analysis. In particular, first and second normal variations are derived this way. The example of the soap film between two rings is generalized to arbitrary variations (and not just axisymmetric ones).