Karsten Groäe-Brauckmann and Konrad Polthier
Abstract
We construct new examples of compact constant mean curvature surfaces numerically. A conjugate surface method allows to explicitly construct examples. We employ the numerical algorithm of Oberknapp and Polthier based on discrete techniques to find area minimizers in the sphere S3 and to conjugate them to surfaces of constant mean curvature in R3. We compute examples of genus 5 and 30 and discuss a further example of genus 3.
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Introduction
For a long time the sphere was the only known compact immersed surface of constant mean curvature 1 (MC1). By the result of Alexandrov there is no other embedded compact MC1 surface and by a theorem of Hopf the only way to immerse the sphere with MC1 is the round sphere. Nevertheless Wente discovered MC1 tori in 1986 [] (see Figure and ) and his work became the starting point for an intensive study of MC1 surfaces.
Pinkall and Sterling classified all tori [] by the genus of a hyperelliptic Riemann surface. This genus should not be confused with the genus of the MC1 surface, which is of course 1 for a torus. Bobenko then found explicit formulas in terms of theta functions associated to the hyperelliptic Riemann surface []. The two papers actually consider a larger class of MC1 surfaces containing the tori. This class gives rise to a period problem for the subset of tori. Algebraic conditions for the solvability of the period problem were stated in [], and Ercolani, Knärrer, Trubowitz [] (also Jaggi []) proved that for each genus there are hyperelliptic curves such that these conditions are satisfied. Hence there are MC1 tori for every hyperelliptic genus. Heil implemented Bobenko's formulas and can solve the period problem numerically [].
Kapouleas constructed compact MC1 surfaces of every genus greater than two by an implicit function argument []. Each surface is based on a graph such that the vertices relate to spheres and the edges to pieces of Delaunay surfaces. Provided the graph is balanced, Kapouleas proves there exists a surface for a suitable scaling of the graph. This means that the edges contain a possibly large number of Delaunay `bubbles'; furthermore the Delaunay handles must be thin. Since the number of bubbles as well as the thinness of the handles are the result of delicate estimates it is almost impossible to decide for a given surface whether it can be obtained by Kapouleas' method. This situation is somewhat similar to the tori for which it seems that the period problem is more likely to be solvable when the surfaces are larger. On the other hand, once the handle size and Delaunay piece length are known Kapouleas' surfaces are very explicit in that they are close to the union of the Delaunay pieces. In [] Kapouleas manages to glue g ä I I N Wente tori together at a single lobe. This yields compact MC1 surfaces of every genus, in particular the up to then open genus 2 case. The handles used are fundamentally different from those of the Delaunay-like surfaces. Kapouleas' Wente-like surfaces fuse tori with a large number of lobes; again the number is practically unknown.
For the first time a conjugate surface method was used by Lawson 1970 [] to prove existence of MC1 surfaces and later extended by Karcher [] and one of the authors []. They constructed a number of periodic MC1 surfaces and surfaces with ends, as well as the Wente torus []. The conjugate surface method is more explicit than Kapouleas' method and works for sufficiently symmetric surfaces.
In the present work we use this method to construct numerically new compact MC1 surfaces of higher genus. As candidates we take highly symmetric MC1 surfaces `close' to a collection of spheres which are joint by small handles. Existence of the Penta surface of genus 5 and the surface with icosahedral symmetry of genus 30 is discussed in detail, while the numerics could not decide an example of genus 3. With any of the methods mentioned before a hard problem is to solve the period problem. In simpler cases it can be settled by an intermediate value theorem [] using a graph property, or by a degree argument []. Here we use the numerical algorithm of Oberknapp and Polthier [] [] to deal with the period problem in more involved situations. The algorithm uses discrete techniques generalizing an algorithm of Pinkall and Polthier [] for minimal surfaces.
At first we review in Section the conjugate surface construction for MC1 surfaces, in its standard C2 description as well as in the C1 form that is used for the numerical algorithm. We also explain the period problem that makes our examples unique (or isolated) for fixed symmetries. We then review the underlying discrete numerical algorithm in Section . In Section we discuss a class of possible candidates, for which the necessary conditions of stability and balancing are satisfied. Three specific examples and their numerics are described in Section in greater detail. In a future paper we want to explore further examples [].
The algorithms and the graphics were implemented using the mathematical programming environment Grape developed at the Sonderforschungsbereich 256 at the University of Bonn. Grape can be obtained on request via the second author.