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**Extra resources for Geometry V. Minimal Surfaces**

**Sample text**

1 above that index(ld) = 1. 2. Montiel and Ros conjectured that it was possible to have a hyperelliptic Riemann surface whose two-fold covering map to S2 had index equal to one, provided the branch values were suficiently well distributed. This was proved by Souam [69]. 3. Given E > 0, choose P = {PI . p,,} c S2 such that the open E-disks about the pi are disjoint, but maximal in the sense that their complement does not contain an E-disk. SupposeC is a hyperelliptic Riemann surface and 4: C + S2 a two-fold covering of S2 whose branch values are precisely P.

Hoffman and H. Karcher 70 The remaining integral s u“+‘dz/z can be rewritten as a linear combination of the integrals we already have. 22) to get (; - z)u k+ldz = (z + &)udz + (x-’ N (k - l)(a + z-‘)zdu + (x-’ 2Re izdu = -$(x+x-‘) I Qa(m) =c. I0 71 -iE + (cot a: - tana) . rn. m. rn. (Ic - l)E-(G)] Then The compatibility and of the form - z)u’“+‘dz. % = E- (al/E+ (a) E- (4/E+(G) E+(G) E+(a) This simplifies to Ly&os2 q5- cos2a cos((-1 + 2/lc)d)d$. 26) This reduction to the same integral as in the lemma allows us to finish the period discussion.

Am. J. Math. 98 (2), 515-528 (1976), Zbl. 53006 2. : Minimal surfacesin R3. Lecture Notes Math. Springer-Verlag 1986,Zbl. ), compiled by meansof the MATH database,and Jahrbuch iiber die Fortschritte der Mathematik (Jbuch) have, asfar aspossible,been included in this bibliography. D. Hoffman 90 and H. Karcher 3. : Elliptische Funktionen und vollst%ndige Minimalflgchen. PhD thesis, FU Berlin, 1989, Zbl. 53007 4. : Deformation families of complete minimal surfaces with four catenoid ends. (Video tape) GANG, August 1991 5.