Higher-genus Chen-Gackstatter surfaces and the Weierstrass representation for surfaces of infinite genus. (English) Zbl 0855.53006

Thayer, Edward C.

Exp. Math. 4, No. 1, 29-39 (1995).

In 1982 B.-Y. Chen and F. Gackstatter constructed two complete minimal surfaces of finite total curvature with genera one and two, each having one Enneper-type end and all the symmetries of Enneper’s surface. The present author combines extensions of Karcher for examples with higher symmetry order and of Chen and Gackstatter for a genus 3 example, and produces the Weierstrass data for a countable collection of surfaces \(M_{p, k}\), where \(p\geq 0\) and \(k\geq 2\) are integers. \(M_{p, k}\) has genus \(p(k- 1)\) and the symmetry of a \(k\)-order Enneper surface and thus is, roughly speaking, an Enneper surface in which \(p\) handles of order \(k\) are inserted along a vertical axis.
With the increasing number of handles the period problem becomes increasingly complex. For \(p\in \{1, 2\}\) the period problems can be solved by intermediate value arguments, but for \(p> 2\) numerical methods must be employed. The present author succeeds in solving the period problem numerically with a simplex method for up to \(p= 34\) handles. He shows that his surfaces \(M_{p, k}\) approximate generalized Scherk towers of order \(k\) for the limit \(p\to \infty\).

Reviewer: K.Polthier (Berlin)

MSC:

53A10

Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

Weierstrass representation; minimal surfaces; finite total curvature; period problem

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