Applications of complex function theory to minimal surfaces

Gregory Scott Rhoads, Purdue University

Abstract

A surface in Euclidean 3-space which is given in terms of isothermal coordinates in a domain contained in the complex plane is a meromorphic minimal surface if each coordinate function is harmonic except for poles. Meromorphic functions can be represented as planar minimal surfaces, thus the theory of minimal surfaces can be considered as a generalization of the theory of meromorphic functions. In this dissertation, we apply complex function theory to obtain results for these surfaces. Beckenbach applied Nevanlinna theory to minimal surfaces and generalized some of Nevanlinna's theorems to these surfaces. In Chapter 1 this application is continued with proving the lemma of the logarithmic derivative. As an application of the logarithmic derivative, the order of a minimal surface is shown to equal the maximum of the orders of the Weierstrass representation functions. After giving some examples of entire minimal surfaces with maximum deficiency sum 2, it is shown that the order of one of these surfaces is a positive integer or infinite which parallels the similar theorem for analytic functions. The examples of deficiency sum 2 are then modified so that one and two deficient values can be arbitrarily chosen. In Chapter 2 the classical differential inequality $\Delta u >f(u)$ where f is a real-valued function is investigated. Conditions on f which imply the nonexistence of twice-continuously differentiable solutions defined in the whole complex plane have been studied extensively. Here, in the case $f(u)=u\sp{1+\epsilon}$, we show no function, u(z), can satisfy the inequality on the set where $u>0$. This result is used to show that for a parabolic surface, the supremum of the curvature is 0 over the set where the metric is large. An analytic function can be locally represented by a power series, and differences in successive exponents in this representation are called gaps. A minimal surface also has a local power series representation, and in Chapter 3 restrictions on its gaps are investigated. Specifically, it is proved that the gaps in successive exponents cannot be strictly increasing, and some results are obtained for the situation where the gaps increase for large exponents only.

Degree

Ph.D.

Advisors

Weitsman, Purdue University.

Subject Area

Mathematics

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