Karsten Groäe-Brauckmann and Konrad Polthier
We describe numerical experiments that suggest the existence of compact constant mean curvature surfaces. Our surfaces come in three families with the genus ranging from 3 to 5, 7 to 10, and 3 to 9, respectively; there are further surfaces with the symmetry of the Platonic polyhedra and genera 6, 12, and 30. We use the numerical algorithm of Oberknapp and the second author that is based on a discrete version of the conjugate surface method.
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A difficult problem in physics is to determine the shapes fluid interfaces make when they divide regions containing other fluids or gases. If an interface is subject to surface tension alone, i.e. tends to minimize its area, then it must have constant mean curvature H. In physical dimensions H represents the difference in pressure between the two sides of the interface. A well-known example are soap films enclosing one or more volumes of air. The corresponding variational problem is area with a volume constraint, and its Euler equation H = const. is a nonlinear elliptic partial differential equation. In the present paper we investigate examples of complete and smoothly immersed surfaces of constant mean curvature. Clearly only an embedded piece of such a surface that is a stable minimum of the variational problem is a soap bubble in the sense that it can arise as a physical interface.
If the mean curvature vanishes, that is in the case of minimal surfaces, there are no complete compact examples, and we exclude this case from now on. We can scale a surface such that a non-zero constant mean curvature is 1 and we adopt the abbreviation MC1 to stand for mean curvature 1. The unit sphere is the obvious example of a compact MC1 surface. It is unique assuming one more property: the sphere is the only embedded compact MC1 surface by the theorem of Alexandrov; and the only immersed topological sphere with MC1 by Hopf's result; furthermore only the sphere is stable for area under a volume constraint (Barbosa and Do Carmo).
The sphere is also the unique compact MC1 surface of revolution. Delaunay solved the respective ordinary differential equation to determine the non-compact MC1 surfaces of revolution in 1841. Such surfaces are simply periodic. The family of solutions includes the embedded unduloids and the immersed nodoids. It is only in the last 25 years that a rapid development in the knowledge of complete MC1 surfaces has taken place. It started with the two doubly periodic surfaces Lawson found in 1970 . In view of the results of Alexandrov and Hopf it did not seem likely that non-spherical compact MC1 surfaces would exist; nevertheless Wente discovered immersed MC1 tori in 1984. Wente's work then stimulated a thorough study of all tori. Pinkall and Sterling consider generalized MC1 tori  and Bobenko  gives formulae for the Riemannian metric in terms of theta functions. If the periods of the surfaces vanish this method leads to compact MC1 tori; it is known that the period condition is in fact satisfied in some cases  . In his numerical work Heil evaluates the theta functions and studies the period problem .
Karcher produced doubly and triply periodic MC1 surfaces with an improvement of Lawson's conjugate surface method in 1989 . More recently a further extension was given by the first author and resulted in surfaces with ends, and further periodic surfaces .
A wealth of surfaces was constructed by Kapouleas, in particular infinitely many compact MC1 surfaces of every genus greater than two . Kapouleas takes initial surfaces composed of segments of Delaunay surfaces glued onto spheres. An implicit function theorem argument is used to prove existence of close-by smooth MC1 surfaces. Nevertheless the surfaces are not explicit: the segments of the Delaunay surfaces must be long, and the handle size tiny - how long and how thin exactly is the result of delicate estimates and hence practically not accessible (see Section for the precise existence statement). Examples of periodic MC1 surfaces and surfaces with ends show that the assumption on the handle size can be considered technical  . In the compact case the present work indicates that the same holds, and applies to the Delaunay segment length as well. - In  surfaces of every genus g ä 2 are constructed in a somewhat similar way using g Wente tori that are glued together at a single lobe. These surfaces are very different from Kapouleas' Delaunay-like surfaces, in that new types of handles are used. Again, Kapouleas chooses one parameter extreme and is in some sense close to the boundary of the moduli space: the fused tori have a large number of lobes.
We would also like to mention the generalized Weierstraß representation of Dorfmeister, Pedit, and Wu . This representation could become an efficient tool to construct MC1 surfaces with specific properties.
In the present paper we give numerical examples of low genus compact MC1 surfaces. Our `planar' examples come in three finite families ranging over different genera; there are further `polyhedral' surfaces. These surfaces are distinguished from other (possible) MC1 surfaces of the same genus by three properties: they are very symmetric; they consist of Delaunay-like pieces; furthermore they are small, that is they contain only few `bubbles', namely g+1 bubbles for a given genus g and fewer for the polyhedral examples. We would like to mention that in addition to the handles used by Kapouleas we also have n-noidal handles that were described in  for the first time.
The numerical existence procedure is based Lawson's conjugate surface method for which symmetries play an essential role. The examples are obtained by planar reflection from a simply connected fundamental domain. All our patches are bounded by five planar arcs and depend on two parameters. On the other hand there are two period conditions to satisfy. Therefore our compact MC1 surfaces are isolated in the symmetric class we consider. To close the periods in a rigorous way one would have to give a loop in the parameter space so that the periods on the loop can be estimated to have nonzero winding number about the origin. Then continuity of the family in its parameters would imply the existence of a surface with 0 periods. Continuity of the surfaces is indeed experimentally observed; it is, however, difficult to prove. The other steps in the existence program can, at least in principle, be proved as in .
Which considerations guided us to look at the MC1 surfaces we present, and which further surfaces can be expected to exist? There are necessary conditions to satisfy: most important is R. Kusner's balancing formula discussed in Section . This condition allows us to reduce the idea for a MC1 surface to a piecewise linear graph that, besides being a topological retract of the surface, also carries essential geometric information (Section ). We consider the simplest possible graphs in this paper. The knowledge of prototypical surfaces (Section and ) indicates that further constraints than those given by the balancing formula are present; in particular it leads to an explanation why our families must be finite.
In a previous paper we constructed three numerical examples of compact MC1 surfaces . These surfaces were chosen close to the degenerate spherical situation, that is with small handles, so that existence could be expected, but not predicted, from Kapouleas' work. Most surfaces we present here have much bigger handles and this makes them numerically easier to deal with. We now believe, our difficulties in obtaining reliable results for our previous genus 3 example are related to numerical problems arising from a surface whose Gauä curvature varies too much.
We use the numerical algorithm of Oberknapp and Polthier   to construct our surfaces. This algorithm generalizes the discrete algorithm of Pinkall and Polthier for minimal surfaces . The algorithm has two steps, one is to minimize area (in fact energy) in S3, the other is to conjugate the surface to a MC1 surface in R3. The algorithm is implemented in the framework of the graphical environment Grape developed at the Sonderforschungsbereich 256 of the University of Bonn.
We can not estimate how close the results of the algorithm are to smooth MC1 surfaces since the algorithm works with discrete data. We are confident they represent actual MC1 surfaces since known MC1 surfaces are reproduced very well with the algorithm. For the compact surfaces dealt with in this paper the period problem is central. We took care to choose our triangulations fine enough and adapted to the curvature as to make sure that the periods obtained for further refinements only vary in a range much smaller than the range of parameters for which the boundary contours exist. Still, care is appropriate as to the exact shape of the surfaces as well as to the exact range of genera of existence, and we hope that future proofs will give a definitive answer to the existence problem.