Branch Points

Abstract

This last chapter contains a new approach to the celebrated result that a minimizer of area in a given contour has no interior branch points. The novelty of the method consists particularly in the fact that, in certain cases, relative minimizers of Dirichlet’s integral are shown to be free of nonexceptional branch points, and this is achieved by a purely analytical reasoning.

Then, boundary branch points of a minimal surface in the class $\mathcal{C}(\Gamma)$ of admissible surfaces with a smooth boundary contour Γ are studied. In particular it is shown that a minimal surface $\mathcal{C}(\Gamma)$ cannot be a minimizer of D in $\mathcal{C}(\Gamma)$ if it has a boundary branch point whose order n and index m satisfy the condition 2m−2<3n (Wienholtz’s theorem).

Furthermore, geometric conditions are exhibited which furnish bounds for the index of interior and boundary branch points. These estimates supplement the bounds on the order of branch points provided by the Gauss–Bonnet theorem.

A special role in all of this is played by the forced Jacobi fields discovered by Böhme and Tromba, which will again show up in the global theory presented in Vol. 3.