The First Main Theorem; Non-exceptional Branch Points; The Non-vanishing of the L ^th Derivative of Dirichlet’s Energy

Abstract

Let us state our main goal: Assuming that is a nonplanar minimal surface in normal form having w=0 as a branch point of order n and index m, we want to show that $\hat{X}$ cannot be a weak relative minimizer of Dirichlet’s integral D in the class . Unfortunately this goal cannot be achieved for all branch points but only for non-exceptional ones and special kinds of exceptional ones. In this chapter we investigate the non-exceptional branch points, while in Chaps. 5 and 6 we deal with the exceptional ones. The main result of the present section – our First Main Theorem – is the following