A Theory of Branched Minimal Surfaces pp 37-45 | Cite as

# Very Special Case; The Theorem for *n*+1 Even and *m*+1 Odd

Chapter

## Abstract

In this chapter we want to show that *a* (*nonplanar*) *weak relative minimizer*\(\hat{X}\)*of Dirichlet’s integral**D**that is given in the normal form cannot have**w*=0 *as a branch point if its order**n**is odd and its index**m**is even*. Note that such a branch point is not exceptional since *n*+1 cannot be a divisor of *m*+1. We shall give the proof only under the assumptions *n*≥3 since *n*=1 is easily dealt with by a method presented in the next section. (Moreover it would suffice to treat the case *m*≥6 since 2*m*−2<3*n* is already treated by the Wienholtz theorem. So 2*m*≥3*n*+2≥11, i.e. *m*≥6 since *m* is even.)

## Keywords

Normal Form Harmonic Function Minimal Surface Branch Point Order Derivative
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© Springer-Verlag Berlin Heidelberg 2012