On the branch point index of minimal surfaces

Let Gamma be a closed, sufficiently smooth Jordan curve in R(3) and denote by C(Gamma) the class of disk-type surfaces X is an element of H(1,2) (B, R(3)) which map partial derivative B continuously and monotonically onto Gamma. Then any minimal surface X is an element of C(Gamma) possesses only finitely many branch points in (B) over bar, and the order of any such point is well-defined, and also the index of an interior branch point is defined in a natural way if X is nonplanar. We show that also the index of boundary branch points can be defined if the curvature kappa and the torsion tau of Gamma are strictly nonzero. Secondly we derive upper bounds for the index of any branch point in terms of the total curvature of Gamma or of its cut number.