Characterization Of The Minimal Surfaces By The Harmonic Functions
The aim of this thesis is to represent a minimal surface in isotermal parameters by harmonic functions and to give distortion inequalities of Weierstrass-Enneper parameters and Gaussian curvature of a certain type of minimal surface.
This thesis consists of five parts. In the first part historical knowledge of the investigation of minimal surface and harmonic functions is presented.
The second part includes eight sections. In Section 2.1., the fundamental concepts of a minimal surface in are given. In Section 2.2., the fundamental definition and theorems of the theory of complex functions are given. In section 2.3., minimal surfaces in isotermal parameters are investigated. Weierstrass-Enneper representation of a minimal surface is given in Section 2.4. and the well-known examples of minimal surfaces are represented in Section 2.5. In Section 2.6., the Gaussian curvature of a minimal surface is investigated. Main part of this thesis is given in Section 2.7. In this section, the necessary and sufficient condition for a minimal surface can be represented by harmonic function is given. In Section 2.8., some classes of harmonic functions which are used to arrive our main aim are given. In the third part, it is summarized how the proofs are done with respect to the tools used.
The fourth part contains, the proofs of distortion inequalities of Weierstrass-Enneper parameters and Gaussian curvature of minimal surfaces corresponding to univalent harmonic functions whose analytic part is starlike and convex respectively.
An evaluation of this thesis is placed in the fifth part.
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