The Problem of Plateau

The following sections are included:

• PREFACE

• TABLE OF CONTENTS

Plateau, Joseph–Antoine–Ferdinand, was born in Brussels on the 14th October 1801. Raised in an artistic environment (his father, a native of Tournai, had a remarkable talent for painting flowers), young Joseph was able to read at the age of six. He made rapid progress in primary school. One day, after having attended a session of "physique amusante", popular in the nineteenth century, he promised himself that sooner or later, he would try to unveil the secret of many mysterious facts which he had observed. He was sent to the Academy of fine arts by his father, evidently to continue the artistic tradition of his family. Unfortunately, he soon became an orphan, as he lost his mother when he was only thirteen, and his father at the age of fourteen. Together with his two sisters, he was taken in by his maternal uncle, master Thirion, a lawyer. After a serious illness, Joseph resumed the courses at the Academy. During the day, he attended with great success the lectures of an exquisite teacher named Van der Meulen, and in the evening, he diverted himself by carrying out experiments of entertaining physics. He constructed all his apparatus with his own hands and often surprised the audience, both by his amazing skill and the originality of his instruments …

This paper begins with a sketch of Radó's life, largely based on the author's recollections. It then traces some main ideas in Radós's work on complex analysis, Plateau's problem, continuous transformations and surface area. A list of Radó's publications is included at the end.

During my three years at Rice University in Houston, Texas, I studied the theory of analytic functions and subharmonic functions as they related to the classical minimal surfaces. My advisor was Professor E.F. Beckenbach who had studied similar topics, as a National Research Fellow, under Professor T. Radó. When I received my Ph.D. in 1940, Beckenbach told me that he had arranged for me to do some post–doctoral work at Ohio State University where Radó was the "reigning" research mathematician. I assumed that I would continue my study of minimal surfaces and analytic functions of one complex variable …

I entered C.C.N.Y. in January, 1961, as a freshman student who was poor in mathematics and physics. In those days you were, as a student, forced to take a certain core group of mathematics courses, along with many others (140 credits needed to graduate with B.S.) …

It was my good fortune to have had Jesse Douglas as a teacher (summer 1941, Columbia University, in Differential Equations), to have shared an office with him at Columbia (General Studies Division) in the early 1950s, and to have been his colleague in the City College (CUNY) mathematics department where he taught for the last ten years of his life, 1955 1965. And my wife, when a student at City College (I did not know her then, as she avoided courses with reputed "tough" teachers) was Douglas' student in his 1961 Modern Algebra course. Thus while not particularly his personal or mathematical confidant, I can yet contribute a few first-hand and some second-hand observations and related items …

Jesse Douglas became a member of the mathematics department at MIT in 1930. He was, at 33 years of age, already a well–known scientist who had written an interesting doctor's dissertation in differential geometry under his teacher Edward Kasner at Columbia University in New York, had studied in Chicago, Paris and Göttingen, and, above all, had already been publishing on his solution of the problem of Plateau, subtle and highly original work for which he would receive the Fields Medal at the Oslo International Mathematical Congress in 1936 …

The exposition commences with a brief summary of Joseph Plateau's 19th century experiments which led to the well-known soap bubble demonstration of minimal surfaces on particular wire frames. This is followed by a brief account of the establishment of the Nobel Prize and the Fields Medal and the award of the latter in its first year of existence to Jesse Douglas for his mathematical proof of what was then the Minimal Surface Problem. The essay concludes with a personal response to the question: Why is there no Nobel Prize for mathematics?

The following sections are included:

• Introduction

• Spherical 2-type hypersurfaces

• Hypersurfaces with a characteristic eigenvector field

• Hypersurfaces satisfying the condition Δx = Ax + B

• References

We study affine minimal surfaces in the 3-dimensional affine space. We completely classify the affine minimal surfaces which have higher order parallel cubic form. We show that they are affine flat and either parallel (in which case we obtain the paraboloids), or affine equivalent to z = xy + P(y), where P is an arbitrary polynomial in y.

Cartan's method of moving frames is used to describe minimal surfaces where curvature, torsion and group properties are considered.

We use a cohomological minimax theorem of Chang to prove multiplicity results for the periodic solutions of some Hamiltonian systems of the second and or the first order with nonlinearity verifying some periodicity condition. Applications are given to systems of coupled pendulums, discretization of Josephson equations and extensions of the Conley-Zehnder results about Arnold's conjecture.

It is very likely that spherical caps are the only compact embedded hypersurfaces with non-zero constant mean curvature in Rⁿ⁺¹ bounded by a round (n - 1)-sphere. Under certain additional assumptions on the considered hypersurfaces, the above conjecture is proved by using the reflection methods of A. D. Alexandrov.

The evolution of the theory of minimal surfaces had two culmination points, one in the publications of general representation formulas relating to complex analysis, by Enneper (1864), Weierstrass (1866), and Riemann (1868, posthumous), and the second in the solution of Plateau's problem for general Jordan curves, by Garnier (1928), Radó (1930), Douglas (1931), and McShane (1933). In our paper we consider basic ideas and aspects in the development of minimal surface theory, particularly emphasizing those that were essential to the solution by Radó. This is based on extensive discussions between Prof. Tibor Radó and the author from 1956 until 1962 at The Ohio State University.

We prove a sufficient criterion for a collection of k-planes passing through a common point to be area-minimizing. The criterion is similar to, but not as sharp as, the "angle criterion," a necessary and sufficient condition for a pair of oriented k-planes to be area-minimizing. The main results are contained in Theorem 8.3 and Corollaries 8.4 and 8.5.

After a general discussion of the subject, we give an exposition of some quantitative results which have been obtained in the last several years. It includes an account of the author's recent results. We also determine the index of Hoffman-Meeks' minimal surfaces with three ends when the genus is not very large.

The following problems in higher-dimensional field theory are considered: the removability of isolated and accumulation-point singularities from low-energy gauge fields; L^p conditions for removing certain classes of submanifold singularities from higher-dimensional gauge fields; extensions to harmonic maps (nonlinear sigma models) and to Yang-Mills fields coupled to matter fields; an a priori inequality for a relevant class of elliptic subsolutions.

We study the second variation of the Willmore functional for surfaces with umbilics using the second variation of area for the conformal Gauss map. A type of Morse Smale index theorem for Willmore surfaces is proved.

Methods of point-set topology are used to construct an example of a homeomorphism from the unit interval onto a subset of R^m having positive m-dimensional Lebesgue measure. This generalizes an example of W. F. Osgood.

It is our object to discuss a few of the open problems on the eigenvalues of the Laplacian, on the number and characterization of minimal surfaces spanning a curve in R³, on the Riemann mapping theorem from the point of view of minimal surfaces as well as on the analogous surfaces in spacetime, in particular manifolds with Lorentz metric (such surfaces are called maximal surfaces). Most of these problems are well known and some of these problems have resisted solution for a long time.

In this survey article we will summarize recent results concerning the structure of compact connected surfaces M⊂R³ with boundary a simple closed planar curve γ lying in the horizontal xy-plane . We assume M is transverse to along γ and we suppose M has non-vanishing mean curvature vector. When M has certain prescribed mean curvature we establish the structure of M with respect to . The foremost interest is the constant mean curvature configurations. We also briefly discuss Bryant Weingarten configurations and extensions to hyperbolic three space.

The paper gives an overview of the development of several areas of research associated with minimal surfaces and surfaces of prescribed mean curvature in which substantial progress has been made in recent years. A large part is devoted to questions related to unstable surfaces, in particular different methods and their relationships are discussed, and some open problems are described.

The following sections are included:

• Introduction

• Embedded Surfaces

• A Paramertisation of the Embedding

• Gravitaion

• String Theory

• Extrinsic Gravity

• ADM Variables

• 1+3 Splitting of the Hilbert Lagrangian

• 1+3 Splitting of the Hilbert Lagrangian in the Normal Gauge

• Conclusions

• References

We show, by a Baire Category argument applied to the parameter space of all minimal immersions between spheres, that linearly rigid minimal immersions abound for sufficiently high degree.