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A Comprehensive Overview of Minimal-Surface Theory

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A Comprehensive Overview of Minimal-Surface Theory A COMPREHENSIVE OVERVIEW OF MINIMAL-SURFACE THEORY, WITH A THOROUGH MATHEMATICAL TREATMENT OF BERNSTEIN'S THEOREM, THE WEIERSTRASS-ENNEPER REPRESENTATIONS, AND PROPERTIES OF THE GAUSS MAP A Thesis Presented to the Faculty of the Department of Mathematics University of Houston In Partial Fulfillment of the Requirements for the Degree Master of Science By Gregory Patrick Ozbolt August 2014 A COMPREHENSIVE OVERVIEW OF MINIMAL-SURFACE THEORY, WITH A THOROUGH MATHEMATICAL TREATMENT OF BERNSTEIN'S THEOREM, THE WEIERSTRASS-ENNEPER REPRESENTATIONS, AND PROPERTIES OF THE GAUSS MAP Gregory Patrick Ozbolt APPROVED: Dr. Min Ru, Chairman Dr. Shanyu Ji Dr. Qianmei (May) Feng Dean, College of Natural Sciences and Mathematics ii Acknowledgement I owe a debt of gratitude to Dr. Min Ru for his guidance and encouragement through- out this endeavor. During my years as an undergraduate student, it was Dr. Ru who helped instill in me a deep appreciation for the power and elegance of differential geometry, a branch of mathematics that soon became one of my favorites. It seemed only natural to base my master's thesis on a topic in differential geometry, so after consulting with Dr. Ru, I decided to base it on minimal-surface theory. By devoting myself to this topic and completing this project, I now have a clear understanding of this rich theory and an even greater appreciation for differential geometry, and I have Dr. Ru to thank for this. His expertise on the subject has been invaluable to me. iii A COMPREHENSIVE OVERVIEW OF MINIMAL-SURFACE THEORY, WITH A THOROUGH MATHEMATICAL TREATMENT OF BERNSTEIN'S THEOREM, THE WEIERSTRASS-ENNEPER REPRESENTATIONS, AND PROPERTIES OF THE GAUSS MAP An Abstract of a Thesis Presented to the Faculty of the Department of Mathematics University of Houston In Partial Fulfillment of the Requirements for the Degree Master of Science By Gregory Patrick Ozbolt August 2014 iv Abstract Notable results in minimal-surface theory include Bernstein's Theorem, the Weierstrass- Enneper representations, and properties of the Gauss map of a minimal surface. For a solution f(x1; x2) of the minimal-surface equation on the whole x1; x2-plane, Bern- stein's Theorem guarantees the existence of a nonsingular linear transformation given by x1 = u1, x2 = au1 + bu2, b > 0, such that (u1; u2) are (global) isothermal param- eters for the surface defined by xk = fk(x1; x2), k = 3; :::; n. We prove Bernstein's Theorem, and we state and prove three important corollaries of Bernstein's Theorem. The Weierstrass-Enneper representations indicate that a minimal surface is defined by the parametrization ~x(z) = (x1(z); x2(z); x3(z)), where the coordinate functions x1(z), x2(z), and x3(z) are expressed in terms of a holomorphic function f and a meromorphic function g in the first case, and in terms of τ = u + iv and a holo- morphic function F (τ) in the second case. We construct the Weierstrass-Enneper representations. Finally, we state and prove a number of results regarding the Gauss map of a minimal surface in E3. For instance, we prove that if M is a complete regular minimal surface in E3, then either M is a plane or else the image of M under the Gauss map is everywhere dense in the unit sphere; we prove that the Gauss map of a complete non-flat regular minimal surface in E3 can omit at most four points of the sphere. Pertinent results in differential geometry and minimal-surface theory have been included throughout this thesis in an effort to make it fairly self-contained. We conclude that the theory of minimal surfaces has broadened into a rich field of study with exciting results that are often unrelated to notions of area from which the theory originally arose. v Contents 1 Introduction. 1 2 Defining a Minimal Surface . 3 3 Classical Examples of Minimal Surfaces. 13 4 The Minimization of Area. 18 5 The Minimal-Surface Equation. 23 6 Isothermal Parameters . 28 7 Bernstein's Theorem . 33 8 The Weierstrass-Enneper Representations of a Minimal Surface in E3 . 43 9 The Gaussian Curvature of a Minimal Surface in E3 ..........................51 10 The Gauss Map of a Minimal Surface in E3 ...................................55 11 Conclusion . 78 References . 80 vi 1 Introduction Differential geometry has long been regarded as a tremendously important branch of mathematics. By utilizing techniques of differential and integral calculus as well as linear and multilinear algebra to study problems in geometry, the mathematical field of differential geometry has proven to be a powerful tool in subjects ranging from topology to Einstein's General Theory of Relativity. A significant discovery that was made in the field of differential geometry was the concept of a minimal surface. The mathematical definition of a minimal surface is a surface whose mean curva- ture is everywhere zero. The meaning of a surface's mean curvature will be developed in the next section. A minimal surface is said to be a surface that minimizes area. In the preface to the book, Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics 2001 Summer School [3], David Hoffman writes that a minimal surface \has the defining property that every sufficiently small piece of it (small enough, say, to be a graph over some plane) is the surface of least area among all surfaces with the same boundary." Classical examples of minimal surfaces include the plane, which is a trivial example; the catenoid, which is the surface formed by rotating a cate- nary curve about its directrix; the helicoid, which is the surface formed by drawing horizontal rays from the axis of a helix curve to points on the helix curve; and the Enneper surface, which is an intricate, self-intersecting surface that was introduced by Alfred Enneper in connection with minimal-surface theory. Before delving into the mathematics associated with minimal surfaces, we shall briefly review the history of minimal surfaces as explained in Hoffman [3]. The prob- lem of determining the surfaces of rotation that minimize area was introduced by the Swiss mathematician Leonhard Euler in 1744. In that same year, he solved the problem and concluded that the catenoid is the only such surface. In 1755, the 1 nineteen-year-old Italian-born mathematician Joseph-Louis Lagrange became inter- ested in the problem of surfaces that minimize area and began to write letters to Euler. In these letters, Lagrange discussed the problem of finding a graph over a region in the plane, with prescribed boundary values, that was a critical point for the area function. In one of his letters to Euler, Lagrange included a second-order, nonlinear, elliptic equation that he had formulated which solved the problem. This equation would become known as the Euler-Lagrange equation. Although Lagrange included this newly-discovered equation for the solution in his letter to Euler, he did not include any new solutions. It is interesting to note that their correspondence led to the formulation of the calculus of variations, a term coined by Euler in 1766. The helicoid was first discovered by Jean Baptiste Meusnier in 1776. He demon- strated that it was also a solution to the Euler-Lagrange equation. Beyond this discov- ery, he gave an extremely important geometric interpretation of the Euler-Lagrange equation in terms of the vanishing of the average of the principal curvatures of the surface. This quantity became known as the \mean curvature," thanks to the sugges- tion of the French mathematician Sophie Germain. The concept of a minimal surface being a surface whose mean curvature is everywhere zero was born. Since their discovery, minimal surfaces have been the subject of immense interest among mathematicians because they are the surfaces that usually minimize area, a trait that can be very useful in many different fields. Over the past fifteen years, the study of minimal surfaces has become the focus of much mathematical and scientific research, notably in the areas of molecular engineering and materials science due to their anticipated applications in nanotechnology. 2 2 Defining a Minimal Surface In order to study the mathematical developments made in the field of minimal sur- faces, we will need to review several fundamental concepts in differential geometry that will lead us to the definition of a minimal surface. The concepts presented in this section and the next have been drawn from Shifrin [7]. We define a surface in R3 to be a connected subset M ⊂ R3 such that each point in M has a neighborhood that is regularly parametrized. Now, we shall define ~x : U ! M ⊂ R3; (u; v) 7! ~x(u; v) to be a C3 one-to-one function from some open set U ⊂ R2 to the aforementioned 3 ~ connected subset M ⊂ R . We will require that ~xu × ~xv 6= 0, where ~xu and ~xv denote @~x=@u and @~x=@v, respectively. We call ~x a regular parametrization of the surface 3 ~ M ⊂ R , regular because ~xu × ~xv 6= 0. We can think of a surface M as containing infinitely many curves within it. Two very important types of curves on M are the u-curves and the v-curves.A u-curve is obtained by fixing v = v0 and varying u, such that u 7! ~x(u; v0), and a v-curve is obtained by fixing u = u0 and varying v, such that v 7! ~x(u0; v). If we let P = ~x(u0; v0) be a point on M, then ~xu(u0; v0) is tangent to the u- curve and ~xv(u0; v0) is tangent to the v-curve. Assuming M is a regular parametrized surface and ~x is a regular parametrization ~x : U ! M ⊂ R3, then we shall define the tangent plane, or tangent space, of M at P , denoted by TP M, to be the subspace spanned by ~xu and ~xv evaluated at P = ~x(u0; v0).
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