Constant mean curvature surfaces with cylindrical ends

R. Schoen has asked whether the sphere and the cylinder are the only complete (almost) embedded constant mean curvature surfaces with finite absolute total curvature. We propose an infinite family of such surfaces. The existence of examples of this kind is supported by results of computer experiments we carried out using an algorithm developed by Oberknapp and Polthier. The cylinder of radius 1 2 is a surface with constant mean curvature 1 (a cmc surface for short). The cylinder has vanishing Gauss curvature K, and hence finite (indeed, zero) absolute total curvature ∫ |K| dA. It is the simplest example of an unduloid. These are the embedded cmc surfaces of revolution, described by Delaunay in 1841 [2] (see also [3]), which are simply periodic and have as generating curves the roulettes of ellipses (of major axis length 1). There is a one-parameter family of unduloids, depending on the eccentricity of the ellipse. For us it is more convenient to parameterize this family by the necksize n ∈ (0, π], which is the length of the shortest closed geodesic. One extreme case of the family is the cylinder, whose necksize is π. At the other extreme, the necksize tends to 0 and the unduloids degenerate to a chain of unit spheres. Periodicity implies that every unduloid, aside from the cylinder, has infinite absolute total curvature. The Delaunay unduloids play a significant role in the theory of embedded cmc surfaces with finite topology, that is, with finite genus g and a finite number of (necessarily annular) ends k. It is a result of Korevaar, Kusner, and Solomon [11] that each of the k ends is exponentially asymptotic to a Delaunay unduloid. Indeed, their results remain true for the slightly larger class of almost embedded surfaces, which are immersed surfaces whose immersion extends to the interior of the surface (see Section 1). We call any such cmc surface a k-unduloid (of genus g). For k ≤ 2 the only k-unduloids are the sphere and the unduloids themselves [11,17]. More than a decade ago R. Schoen raised the question of whether there are any complete (almost) embedded cmc surfaces with finite absolute total curvature, besides the sphere and cylinder. Such a surface must have finite topology [1]. Thus, by the asymptotics theorem [11], the question is equivalent to the problem we address in the present paper: Problem 1. Can a k-unduloid have all of its ends cylindrical for k ≥ 3? 2 Große-Brauckmann, Kusner, and Sullivan It is worth noting that simply or doubly periodic surfaces with (an infinite number of) cylindrical ends exist [4]. Of course, these all have infinite topology and infinite absolute total curvature, though the absolute total curvature of each (non-compact) fundamental domain is finite. 1 Immersed examples and almost embeddedness Interesting complete, non-compact immersed cmc surfaces of finite absolute total curvature are known: for example, Pinkall and Sterling depict a cmc surface with genus zero and two cylindrical ends [19]. It looks like a “two-lobed Wente torus” fused to a cylinder, and its existence was proven later (see [20], [4], and also [21] where similar surfaces can be constructed for arbitrary Delaunay ends). Since the ends of this surface are embedded they have again exponential decay to a true cylinder, so its absolute total curvature is finite. These (and many other) examples suggest that the class of all immersed cmc surfaces is too large to give much control on their geometry. When studying minimal surfaces, embeddedness is often a natural condition to impose, especially for physically motivated problems. The maximum principle implies that, under continuous deformation, a complete minimal surface cannot suddenly stop being embedded unless self-intersections occur at the ends of the surface. The situation is different in the case of cmc surfaces, as seen in Figure 1: when we continuously deform an embedded cmc surface, embeddedness may be lost as bubbles start to overlap. This leads us to concentrate on a natural class of immersed surfaces which arise when considering families of embedded cmc surfaces, the almost embedded surfaces, mentioned above. By definition, an immersed surface is almost embedded if it can be parametrized by an immersion f : M → R which extends to an immersion F : Ω → R, where Ω is a three-manifold with ∂Ω = M . In fact for cmc surfaces of finite topology, the methods of [13] imply Ω can always be taken to be homeomorphic to a handlebody in R. The principal results on finite topology cmc surfaces are valid for this almost embedded class: for instance, ends are asymptotic to Delaunay unduloids [11], each k-unduloid remains a uniformly bounded distance from a k-ended piecewise-linear graph [10], and the moduli space of all these kunduloids (near a surface with no L Jacobi fields) is a real analytic manifold [14] of dimension 3k − 6. 2 Nonexistence results for cylindrical ends There is evidence that k-unduloids with only cylindrical ends are rare. For example, we have proven that there are no k-unduloids of genus zero with all ends cylindrical when there are only k = 3 ends [6]. More generally, when all the ends have their axes in one common plane — a case we call coplanar — there are at least two non-cylindrical ends provided g = 0 and k is odd. Constant mean curvature surfaces with cylindrical ends 3 Fig. 1. Two triunduloids indicate a continuous transition from embedded to nonembedded cmc surfaces. The second surface, whose bubbles overlap, is still almost embedded. There is further evidence for the rarity of these examples from a different perspective. Gluing constructions were introduced by Kapouleas [8], and have become a powerful and general tool to produce examples of cmc surfaces. Two unduloids, for example, can be glued together by connecting them with a small, almost catenoidal neck. Here, as in general, the resulting surface will have slightly different axis directions of the ends, and it may be necessary to perturb the necksizes, too. These changes can be made arbitrarily small, however, when the connecting neck is small enough. For Kapouleas’ construction to be applicable, the two unduloids must have small necksize, and it is natural to ask if we can similarly glue two tangent cylinders together by a small neck. The two cylinders themselves form a degenerate surface which can be naturally regarded as lying in the boundary of the moduli space of 4-unduloids; in fact there is a one-parameter family of tangent cylinders, parameterized by the angle of their axes. We might expect to find examples with only cylinder ends in the interior of moduli space within a neighborhood of these boundary points. Kapouleas, Mazzeo, and Pollack have recently announced [16, p.7] a gluing construction (inserting a small catenoidal neck between the cylinders) which apparently yields all 4-unduloids in such a neighborhood. However, on each of the surfaces constructed this way, at least two of the ends necessarily decrease their necksize, and thus are no longer cylindrical. This change can be made arbitrarily small when the gluing neck is small, but the change is always present. Similarly, when h cylinders are glued to form a 2h-unduloid, then at least h necksizes must change. Since this type of construction always changes some necksizes, there are no examples with all cylindrical ends near these boundary points of moduli space. 4 Große-Brauckmann, Kusner, and Sullivan 3 The necksize problem Consider a k-unduloid of genus g. Let n1, . . . , nk ∈ (0, π] be the asymptotic necksizes of its ends (the lengths of the shortest closed geodesics on the limiting Delaunay unduloids), and let a1, . . . , ak ∈ S 2 be their (outward oriented) axis directions. These asymptotic quantities satisfy the balancing formula [11]