Boundary Branch Points of Minimal Surfaces Spanning Extreme Polygons

Abstract.

In this paper the author shows the non-existence of boundary branch points of minimal surfaces spanning extreme simple closed polygons in ${\mathbb{R}}^3$. Moreover a minimal surface whose boundary values only map into the straight lines that are determined by two adjacent vertices P _j , P_j+1 of some extreme simple closed polygon Γ and which is required to possess at most one branch point, lying on the boundary $\partial B$, is shown to be already free of branch points. Both of these results are of great importance for the author’s proof of the finiteness of the number of all stable immersed minimal surfaces spanning any fixed extreme simple closed polygon, in [6]. Moreover the significance of the first mentioned theorem is even increased by the fact that the author could generalize his “finiteness theorem” of [6] in his paper [7] to the statement that any simple closed polygon which has the desired property to bound only minimal surfaces without boundary branch points can span only finitely many stable immersed minimal surfaces.