Constant mean curvature surfaces constructed by fusing Wente tori

Closed smooth surfaces of any genus g ? 2, immersed in E3 and of constant mean curvature, are constructed, by "fusing" Wente tori. A fundamental problem in mathematics whose answer has important consequences in many scientific disciplines is the so-called isoperimetric problem, which asks which surface has the least area among the surfaces enclosing a fixed volume. The answer is well known to be the round sphere, although a rigorous proof of this fact requires a certain amount of mathematical ingenuity and sophistication. More generally, one can ask which surfaces have critical area subject to the requirement that they enclose a fixed volume. Such surfaces are often called soap bubbles because a soap film-or more generally a fluid interface-in equilibrium between two areas of different pressure and subject only to the forces induced by this pressure and the surface tension has critical area subject to a fixed enclosed volume constraint. The differential equation characterizing such surfaces locally is H = constant $ 0, where H is the mean curvature. We adopt the abbreviations from now on "CMC surface" to stand for "nonzero constant mean curvature, immersed, smooth, closed surface in E3," and "CMC immersion" for the corresponding immersion. E3 stands for Euclidean threedimensional space (equipped with the usual flat metric), a mathematical model of the space we live in. Since closed surfaces have no minimal immersions into E3, a CMC immersion is from the geometric point of view most appealing among all E3 immersions of the surface. One is led then to the question of which surfaces have CMC immersions and more specifically whether there are any CMC surfaces that are not round spheres. Such questions have been studied seriously by differential geometers for a long time, and in 1853 Jellet proved that star-shaped CMC surfaces are round spheres (1). A century later Hopf established the same for topological CMC spheres (2), andshortly afterwards Alexandrov did the same for embedded CMC surfaces (3). Finally, Barbosa and doCarmo showed that local minimizers of the variational problem are round spheres (4). Because of these results the prevailing mood for a while was that the round spheres are the only CMC surfaces. In 1982 Hsiang demonstrated, however, that this fails, at least in higher dimensions (5). In 1984 Wente in a surprising development achieved the construction of nonspherical CMC surfaces (6). The first examples were tori and the methods used made use of the facts that the Hopf differential, a quadratic holomorphic differential associated to a CMC surface, takes a particularly simple form on a torus and that the commutativity of the fundamental group of the torus forces all of its representations into the Euclidean group to consist of motions sharing a common axis. The Wente construction has been analyzed and extended (7-12), and today we have a classification and a much better understanding of the CMC tori. These methods, however, have failed to The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. ?1734 solely to indicate this fact. produce any results up to now for CMC surfaces of genus g ?2. In 1987 the author announced the construction of CMC surfaces of any genus g 2 3 (13). The methods used were entirely different from the above, and the construction consisted of fusing Delaunay surfaces (14) to obtain surfaces of mean curvature H close to 1, then perturbing the latter to get the desired surfaces with H 1 (13, 15, 16). Delaunay surfaces can be parametrized by a parameter T so that a fundamental domain of the surface consists of the following two pieces: First, a positively curved region that is a perturbation of a round sphere minus two diametrically opposed small (as T -O 0) discs. The perturbation is of order 1X1 on compact sets of the round sphere minus the appropriate antipodal points. Second, a negatively curved region which expanded by a factor 1/h tends to a catenoid as T -O 0. The circles where the curvature changes sign are of order 'P. The fusion involves the spherical pieces. The construction works provided that a certain balancing condition is satisfied at each nearly spherical region where fusion occurs. The CMC surfaces constructed this way are actually Z-coverings of the closed surface, where the period involved is a translation which can be arranged to vanish by appropriately varying the T parameter (16). The balancing condition turns out to prohibit the construction of genus g = 2 CMC surfaces by this construction. The author hopes that the construction above can be generalized to a construction where CMC surfaces-including hypersurfaces in En (n 2 3) besides the usual case of surfaces in E3 (notice that there is nothing intrinsically two-dimensional in this method)-are constructed by using as ingredients complete minimal (hyper)surfaces scaled to be of very small size and with a neighborhood of their infinity removed and round spheres with a number of small discs removed. If these ingredients are appropriately placed and of the appropriate size, one could join them by appropriate surfaces so that a closed surface of mean curvature H close to 1 would be obtained and subsequently perturbed to a CMC surface. There are many difficult obstacles to be overcome before such a general construction can be carried out. Specifying correctly the joining pieces in particular seems especially difficult. The purpose of this paper is to announce a much more modest generalization of the old construction: The old construction is extended by allowing the fusion of Wente tori as well-such a fusion is possible because the Wente tori, like the Delaunay cylinders, contain nearly round-spherical regions. This way the joining pieces above are "borrowed" from the Wente tori, as in the old construction they were "borrowed" from the Delaunay cylinders. The new construction is enough to settle the question of whether all surfaces have CMC immersions, because it allows us to "fuse" g Wente tori for any g 2 2. It turns out that to prove that this construction works, we need a much deeper understanding of the mathematical Abbreviations: CMC, constant mean curvature; asr, almost spherical region.