A long-standing problem in the theory of minimal surfaces is the construction of complete embedded minimal surfaces with finite topology. Many examples of complete properly embedded minimal surfaces in Euclidean three-space have been constructed, but except for the plane, the catenoid, and the helicoid, all the known surfaces have infinite genus. Since it is natural to try to develop a theory of the global geometry of embedded minimal surfaces of finite type, the lack of examples is a major obstacle. In fact, it has often been conjectured that these three examples are the only complete embedded minimal surfaces in R 3 of finite topological type. In 1980, Jorge and Meeks [3] developed a theory to study the topology of complete embedded minimal surfaces in R 3 of finite total curvature. (By a classical theorem of R. Osserman [4], a complete minimal surface of finite total curvature is conformally a compact Riemann surface with a finite number of points removed.) They were able to prove that there were no complete embedded minimal surfaces of finite total curvature of genus zero with three, four, or five ends, and, except for the plane, an embedded complete minimal surface of finite total curvature had at least two ends. Recently, R. Schoen [5] has proved that the only complete embedded minimal surface of any genus with finite total curvature and two ends is the catenoid. The catenoid has genus zero, two ends, and total curvature-Air. We have found that there exist embedded minimal surfaces of finite total curvature of every genus. THEOREM 1. For every genus g > 0 there exists a complete embedded minimal surface M g of genus g with three ends and finite total curvature-4TT(<7 + 2). Using this theorem we have the following corollary. COROLLARY. For every nonnegative integer k, except k = 2, there exists a complete embedded minimal surface with total curvature equal to-4/c7r. PROOF OF COROLLARY. The examples of Theorem 1, together with the plane and the catenoid, give examples of complete embedded minimal surfaces of total curvature G =-4irk for every nonnegative integer k except k = 2. In [3] it is proved that on a complete embedded minimal surface of finite total curvature, G =-4ir(g + r-1), where g is the genus and r > 1 is the number
In a previous paper, new closed embedded smooth minimal surfaces in the round three‐sphere S3()1 were constructed, each resembling two parallel copies of the equatorial two‐sphere Snormaleq2 joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel, we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the linearized doubling (LD) methodology developed in the previous paper. The LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides [15]. In particular, they allow us to develop doubling constructions for the catenoid in Euclidean three‐space, the critical catenoid in the unit ball, and the spherical shrinker of the mean curvature flow. Unlike in the previous paper, our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be “densely distributed,” that is, they do not miss any open set of Snormaleq2 in the limit. This in particular leads to interesting observations that seem to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities by concentrating infinitely many catenoidal necks at a point. © 2019 Wiley Periodicals, Inc.
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 ofdifferent 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), and-shortly 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 ofnonspherical 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)(8)(9)(10)(11)(12), and today we have a classification and a much better understanding of the CMC tori. These methods, however, have failed to produce any results up to now for CMC surfaces...
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