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

Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we want to show that a (nonplanar) weak relative minimizer\(\hat{X}\)of Dirichlet’s integralDthat is given in the normal form cannot havew=0 as a branch point if its ordernis odd and its indexmis 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 2m−2<3n is already treated by the Wienholtz theorem. So 2m≥3n+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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

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