Curvatures, Functionals, Currents

The starting point is the problem of minimizing functionals ℱ(M), defined on surfaces M and depending on the curvatures of M. The point of view is that of the direct method in the Calculus of Variations. The key idea is to consider the graph G of the Gauss map of a surface M, and to consider functionals ℱ such that Area(G) ≤ cℱ(M). Then one can use the known compactness results for integral currents to obtain convergent minimizing sequences of graphs Gj. The currents which are possible limits of such minimizing sequences and their generalized curvatures are investigated. Functionals defined on these classes of currents are studied.