The Boundary Behaviour of Minimal Surfaces

Abstract

This and the next chapter deal with the main topic of Vol. 2, the boundary behaviour of minimal surfaces, with particular emphasis on the behaviour of stationary surfaces at their free boundaries. This and the following chapter will be the most technical and least geometric parts of our lectures. They can be viewed as a section of the regularity theory for nonlinear elliptic systems of partial differential equations. Yet these results are crucial for a rigorous treatment of many geometrical questions, and thus they will again illustrate what role the study of partial differential equations plays in differential geometry. It will be proved that a minimal (or H-) surface spanning a given boundary configuration behaves as smoothly at its boundary as the regularity class of its fixed or free boundary contour indicates. Results of this kind are basic for many investigations. In addition asymptotic expansions for minimal surfaces at boundary branch points and a general version of the Gauss–Bonnet formula is derived.