Embedded minimal surfaces with an infinite number of ends

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Fig . 8. The polygonal boundary Ц,

meeting at Pq have unit length. Since к is fixed in this discussion, the angle о = 7г/(/с+1) at Pq is also fixed, so the rescaled Г is completely determined by the length b oï piP2. Let /q, /i and 12 refer, respectively, to the line segments PoPi and P1P2 and the ray emanating from p2. Note that yi = lf. We label the rescaled boundary curve /^. See Fig. 8.

Proposition 3. There exists a value of b for which the period, defined above, ishes. The conjugate surface S = Df extends by reflection to a minimal surface satisfying all the conditions of Theorem I.

The proof will proceed as follows. We establish first, in Lemma 2, that every Tjj, b>0, is spanned by some minimal disk D^, satisfying all the conditions of Lemma L Therefore, it makes sense to speak of c*(b), as a multivalued function. (Note that c*(b) is different from the function c{t).) In Lemma 3 we show that Dj, converges, as b^O, to a planar region and describe the limiting behavior of the Gauss map of D^, on ^D^, = /^. Lemmas 4 and 5 are needed to establish Lemma 6, which allows us to estimate the twisting of the Gauss map on ^D^, = /^ for large b. After properly orienting the /^ we use Lemma 3 to show that c* (b) > 0 for b small, and then use Lemma 6 to establish that c*(b)<0 for b large. A continuity argument is then given to prove the existence of a Ь with Oec*(b).

This argument is carried out explicitly after Lemma 6, and may be read now assuming Lemmas 2, 3 and 6.

Lemma 2. Every /^ is spanned by a minimal disk Df^ satisfying the conditions of Lemma L

Proof To prove the existence of D^, we begin by fixing b and converting Fj, to a finite contour, Ц, in the following manner. For t>0, truncate the rays at length t and connect their endpoints by a circular arc in the plane spanned by the rays. Then approximate /^ by a sequence {/^1} of smooth extremal curves