## 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

##### 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(x_1, x_2) of the minimal-surface equation on the whole x_1, x_2-plane, Bernstein's Theorem guarantees the existence of a nonsingular linear transformation given by x_1 = u_1, x_2 = au_1 + bu_2, b > 0, such that (u_1, u_2) are (global) isothermal parameters for the surface defined by x_k = f_k(x_1, x_2), 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) = (x_1(z), x_2(z), x_3(z)), where the coordinate functions x_1(z), x_2(z), and x_3(z) are expressed in terms of a holomorphic function f and a meromorphic function g in the first case, and in terms of t = u + iv and a holomorphic function F( t) 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 E^3. For instance, we prove that if M is a complete regular minimal surface in E^3, 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 E^3 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.