By Osserman R. (ed.)
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Additional info for Geometry V. Minimal Surfaces
Thus I = -e2ifl(T)2 r/k-vertex itself; i. e. Since o and p were both chosen in the range (0, $), cx = /3 and uk Note that T = s and that M takes -1 to 0, 0 to i and 1 to oo. Since T takes on the values -l,O, -iepi”, -epicu at the vertices of the fundamental 4-gon, we know that z takes on the values 0 and i at two of the vertices. 6 We note that z extends to the 2k-gon with a zero of order k at the midpoint, and covers the hemisphere k/2 times. Extending to the rest of S by Schwarz reflection produces a function of degree k whose zeros and poles (all of order k) occur at the special polar points.
27). 2. 3 that the Riemann surfacesdetermined by the conformal parameter a, with CE> 7r/4, are obtained from those with a < r/4 by exchanging generators. e. 26) deal with all the terms in to give , D. Hoffman 72 and H. Karcher I. Complete where z = -m-l) on the planar geodesic symmetry curve between the endpunctures at the points where z = --z-l = - tana and z = co. To seethis, let (u > 7r/4. 29) m-l-tana+(cota-m)>(k-l)(cota+tana)>O. This shows that M: = m-l - m > 0, that m = -T + 1/l + y < l/m, and that m < cot a.
M;,T is compact. Specifically, every sequence of surfaces in MG k possesses a subsequence that converges smoothly on compact subsets of R3 to a surface in MG,k. Fig. 3. Five-ended embedded minimal surface of genus 3, discovered by Boix and Wohlgemuth. See (4) in the list above with flat ends) can be proved by theoretical means involving finding desingularizing limits of unstable solutions to the Plateau Problem. 2, this procedure has never produced an example not previously discovered by the method outlined above.