Constant mean curvature surfaces constructed by fusing Wente tori

Abstract

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...

Constant mean curvature surfaces with Delaunay ends

“…Further developments were given in [7]. More recently, using different methods based on the Schwarz reflection principle, Grosse-Brauckmann has constructed families of CMC surfaces with k ends and with fc-fold dihedral symmetry [3].…”

“…The former we have already encountered; on the other hand, a minimal fc-noid is by definition a complete minimal surface S of finite total curvature with k ends. Denote the moduli space of minimal fc-noids of genus g by Hg 7 k-This space (or rather, a very closely associated one) has been studied by Perez and Ros [17], and they prove the result corresponding to that of [9] that this space is real analytic of virtual dimension 3fc. Elements of these spaces have been shown to exist by the classical Weierstrass method, and more recently elements have been constructed for g very large by Kapouleas by a desingularization scheme [6] and a connected sum result, analogous to that in [11], has been obtained by S. D. Yang [21].…”

“…By studying the Cauchy data of the functions producing these normal graphs, and prove that we may match the Cauchy data from the inner piece (the graph over the fc-noid) with that from the ends, thus one advantage of this procedure is that more of the technical complications caused by the nonlinearities are avoided than would be otherwise. Another important issue is that in previous constructions ([5], [7], [21]) one must face the delicate issue of balancing forces, or creating small nonvanishing forces in the approximate configurations. We do not need to deal with this issue here because we implicitly use the balancing of forces already existing in the minimal fc-noids.…”

Construction of compact constant mean curvature hypersurfaces with topology

“…The first modern breakthrough was Wente's discovery, detailed in [20], of immersed constant mean curvature tori. Not long afterwards, Kapouleas used transcendental PDE methods to construct many new constant mean curvature surfaces, including compact ones with arbitrary genus [10], and noncompact ones with finitely many ends [9]. Große-Brauckman in [2] subsequently constructed certain of these noncompact surfaces with large discrete symmetry groups using more classical methods based on Schwarz reflection.…”

Gluing Eguchi‐Hanson Metrics and a Question of Page

“…This method has been used to construct solutions of many nonlinear elliptic equations arising in differential geometry (see e.g. [2], [3], [6], [8], [19], [23], [24], [25], [26], [28], [27], [30], [31], [36], [39], [40]). …”