On Baernstein’s theorem for minimal surfaces

In this chapter we adapt the Riemann-Hilbert boundary value problem, which is a classical complex analytic technique, to minimal surfaces and holomorphic null curves in Euclidean spaces. The Riemann-Hilbert modification method obtained in this way provides a powerful new tool in the theory of minimal surfaces, which is used in the proofs of all main results in subsequent chapters. We also develop a method of exposing boundary points of minimal surfaces, a new tool that is particularly useful in the Calabi-Yau problem and the hitting problem for minimal surfaces.

One-sided surfaces present themselves quite naturally in the theory of minimal surfaces; they were first studied systematically by Sophus Liet under the name of Minimaldoppelflachen. Lie had interpreted the formulas of Monge for a minimal surface in the now well known geometric way: that every minimal surface can be generated by translating one minimal curvet along another. An equivalent construction is to take the mid-points of all line segments whose ends lie respectively on any two fixed minimal curves gi, g2; their locus is a minimal surface, which is real whenever ,ut, A2 are conjugate complex. If li and A2 coincide in a minimal curve A, this construction becomes the taking of the locus of the midpoints of all chords of ,u. The resulting minimal surface, real whenever / is its own conjugate complex curve, is of the type designated as Minimaldoppelflache by Lie. If this surface is not periodic, i.e. does not go over into itself by a certain translation and its repetitions, then it is a one-sided surface in the sense of topology: a,material point moving continuously on the surface can pass from any position to that directly beneath without crossing over any boundary of the surface (M6bius strip); or, if we ascribe a certain arrow to the normal at any fixed point and follow the continuous variation of this sensed normal as the point moves on the surface, it is possible to describe a closed path which will reverse this arrow. In particular, the surface cannot be periodic if it is algebraic; therefore every algebraic double minimal surface is one-sided. Algebraic character can be secured for the surface by taking the minimal curve pu to be algebraic; this in turn can be done by taking as algebraic the arbitrary function f(t) in the Weierstrass formulas for a minimal curve:?

Some generalizations of classical results in the theory of minimal surfaces f: M Rn are shown to hold in the more general case of harmonically immersed surfaces. Introduction. Let (M, g) be a connected Riemann surface with a prescribed metric g in its conformal class and let f: M > Rn be an immersion. It is well known that f realizes M, with the induced metric from Rn, as a minimal surface if and only if f is a conformal (with respect to g) harmonic map (cf., for example, [3 or 8]). That is, the theory of minimal surfaces is substantially the theory of conformally immersed harmonic surfaces. Our purpose is to analyze the case when f is simply a harmonic immersion, to introduce an appropriate Gauss map and, as the main achievement, to establish in this new setting the analogue of three fundamental results (cf. Theorems 1.1 and 2.1, and §3) in the theory of minimal surfaces: the harmonicity of the Gauss map (Ruh and Vilms [13]), equidistribution properties of the Gauss map in cPn (Chern and Osserman [1 and 11]), and the Enneper-Weierstrass representation formulas recently due, for arbitrary codimension, to Hoffman and Osserman [7]. The last two topics (the third for n = 3) have alrealy been treated by T. K. Milnor (see for instance her survey article [9]), but as we remark in §2, our results complement hers in an interesting way. The method of the moving frame as well as the Einstein summation convention are used throughout this paper. 1. The Gauss map and first properties. Let (M, g) be as in the Introduction. We fix the index ranges 1 Rn be an immersion. A Darboux frame al g f is a map E: U c M E(n), U ope in M, such that Received by the editors April 6, 1987. Selected results of this paper were presented by the first named author November 1, 1987 at the 837th meeting of the American Mathematical Society held at Lincoln, Nebraska. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A07, 53A10.

The theory of Minimal Surfaces is one of the most interesting subject in mathematics. The study and computation of minimal surfaces has a long history. Lagrange made the first investigation of the minimal surfaces by asking a simple question named as Plateau’s problem which concerns with finding a surface of least area that spans a given fixed one-dimensional contour in three-dimensional Euclidean space. The study of this kind of surfaces has motivated the development of numerous concepts and techniques in the calculus of variations which later have been applied to solving a great quantity of problems in very diverse areas ranging from the study of partial differential equations to topology and even to settling conjectures in relativity. Mathematically, a minimal surface corresponds to the solution of a nonlinear partial differential equation. In this paper, we define two types of Translation-Factorable (T-F) surfaces in the 3-dimensional hyperbolic space. Then, we obtain the complete classification of these surfaces with vanishing mean curvature by solving some differential equations.

In this chapter we shall discuss two extensions of the theory of minimal surfaces. First we shall solve the problem of finding minimal surfaces of least area when the whole boundary or part of it is not prescribed but left free on given manifolds.1 Secondly we shall study minimal surfaces whose areas are not relative minima. Minimal surfaces of this type correspond to unstable equilibria of a soap film; they will therefore he referred to as unstable minimal surfaces.

Minimal surfaces with constant Gaussian curvature in real space forms have been classifiedcompletely (cf. [Ca-2], [Ke-1], [Br] and their references). Next natural interest is to investigate minimal surfaces with constant Gaussian curvature in complex space forms, more generally in symmetric spaces. Prof. Kenmotsu posed the following problem: Classify minimal surfaces with constant Gaussian curvature in complex space forms. Recently, minimal 2-spheres with constant Gaussian curvature in complex projective spaces were classifiedindependently by [B-Oh] and [B-J-R-W]. [C-Z] studied pseudo-holomorphic curves of constant curvature in complex Grassmann manifolds. For an immersion <p of a Riemann surface M into a Kahler manifold N, the Kahler angle 0 of <pis defined to be the angle between Jd<p{d/dx) and d(p{d/dy), where z=x + V―ly is a local complex coordinate on M and / denotes the complex structure of N. Chern and Wolfson [Ch-W] pointed out the importance of the Kahler angle in the theory of minimal surfaces in Kahler manifolds. In [B-J-R-W] and [E-G-T] they investigated minimal 2-spheres in complex projective spaces and minimal surfaces in 2-dimensional complex space forms respectively in terms of the notion of Kahler angle. In this paper we classifyminimal surfaces with constant Gaussian curvature and constant Kahler angle in complex space forms.

We give a summary of the computer-aided discoveries in minimal surface theory. In the later half of the 20th century, the global properties of complete minimal surfaces of finite total curvature were investigated. Proper embeddedness of a surface is one of the most important properties amongst the global properties. However, before the early 1980s, only the plane and catenoid were known to be properly embedded minimal surfaces of finite total curvature. In 1982, a new example of a complete minimal surface of finite total curvature was found by C.J. Costa. He did not prove its embeddedness, but it was seen to satisfy all known necessary conditions for the surface to be embedded, and D. Hoffman and W. Meeks III later proved that the surface is in fact embedded. Computer graphics was a very useful aid for proving this. In this paper we introduce this interesting story.

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space \(\mathbb {E}^3\) is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space \(\mathbb {L}^3\). In this article, we introduce a new deformation family that continuously connects minimal surfaces in \(\mathbb {E}^3\) and maximal surfaces in \(\mathbb {L}^3\), and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the López-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space \(\mathbb {I}^3\) appear in the middle of the above deformation family. We also prove another type of Krust’s theorem for this family, which implies that the graphness of such minimal surfaces in \(\mathbb {I}^3\) strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory.

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some canonical embedded surface. From the point of view of geometry, a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces. The simplest functional is the area functional. The extremal surface of the area functional is called the minimal surface. Such minimal surfaces were used extensively by Meeks-Yau [MY21 in studying group actions on three dimensional manifolds. In [MY2], the theory of minimal surfaces was used to simplify and strengthen the classical Dehn's lemma, loop theorem and the sphere theorem. In the setting there, one minimizes area among all immersed surfaces and proves that the extremal object is embedded. In this paper, we minimize area among all embedded surfaces isotopic to a fixed embedded surface. In the category of these surfaces, we prove a general existence theorem (Theorem 1). A particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area. We also prove that there exists an embedded sphere of least area enclosing a fake cell, provided the complementary volume is not a standard ball, and provided there exists no embedded one-sided RP2. By making use of the last result, and a cutting and pasting argument, we are able to settle a well-known problem in the theory of three dimensional manifolds. We prove that the covering space of any irreducible orientable three dimensional manifold is irreducible. It is possible to exploit our existence theorem to study

The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.

Minimal surfaces are the surfaces of the smallest area spanned by a given boundary. The equivalent is the definition that it is the surface of vanishing mean curvature. Minimal surface theory is rapidly developed at recent time. Many new examples are constructed and old altered. Minimal area property makes this surface suitable for application in architecture. The main reasons for application are: weight and amount of material are reduced on minimum. Famous architects like Otto Frei created this new trend in architecture. In recent years it becomes possible to enlarge the family of minimal surfaces by constructing new surfaces.

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in H3(−1), and prove that it cannot omit more than four points unless the surface is a horosphere. 0. Introduction In minimal surface theory, the value distribution of the Gauss map has been studied for a long time. An early result is the classical Bernstein Theorem: Any complete minimal graph in R is a plane. Replacing the graph by some geometric conditions, R. Osserman [3] showed that if the complete minimal surface is nonflat, then the Gauss map cannot omit a set of positive logarithmic capacity, which answered a conjecture of Nirenberg. Afterwards, it was generalized by F. Xavier [8] that the Gauss map of such a surface can omit at most six points. In 1989, H. Fujimoto [2] proved the best result finally that the Gauss map of the nonflat complete minimal surface in R can omit at most four points. It is natural to consider the similar properties of minimal surfaces in H. An interesting feature is that there exists a family of absolutely area-minimizing hypersurfaces in H and only one of them is totally geodesic (see [7]). The question seems to be how to raise an adequate Bernstein problem in hyperbolic space. Professor Y. L. Xin pointed out to me that the striking work done by R. Bryant [1] supplies a framework to solve the Bernstein problem in hyperbolic space. In this paper, we shall be concerned with the surfaces in hyperbolic space of constant mean curvature one. We abbreviate “constant mean curvature one” by CMC-1. These surfaces share many properties with minimal surfaces in R. They possess the “Weierstrass representation” in terms of holomorphic data. This formula was discovered by R. Bryant [1]. Many other properties may be found in papers by M. Umhara and K. Yamada ([4], [5]). Here we try to investigate the hyperbolic analogue of the Gauss map. It is a natural question how the values of the hyperbolic Gauss map distribute. Using Bryant’s representation formula we are able to answer this question as follows. Theorem. The hyperbolic Gauss map of nonflat complete CMC-1 surfaces in H3(−1) can omit at most four points. Received by the editors November 1, 1995 and, in revised form, April 2, 1996. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42.

A martingale approach to minimal surfaces

We point out an interesting connection between fluid dynamics and minimal surface theory: When gluing helicoids into a minimal surface, the limit positions of the helicoids correspond to a ‘vortex crystal’, an equilibrium of point vortices in two-dimensional fluid that move together as a rigid body. While vortex crystals have been studied for almost 150 years, the gluing construction of minimal surfaces is relatively new. As a consequence of the connection, we obtain many new minimal surfaces and some new vortex crystals by simply comparing notes.

Nontrivial minimal surfaces in a class of Finsler spheres of vanishing [formula omitted]-curvature