Gauss maps and moduli spaces of minimal surfaces in Euclidean spaces

This work is divided into two parts.;The first part (Chapter 2) is devoted to the study of the value distribution properties of the Gauss map of complete minimal surfaces in m dimensional euclidean space. It is proved that if the Gauss map of such a surface intersects more than m(m + 1)/2 hyperplanes in general position only a finite number of times, the surface must have finite total curvature. This is a generalization of the Picard type theorem for holomorphic curves.;The second part of this paper (Chapter 3) is concerned mainly with the study of moduli spaces of complete minimal surfaces in 3 dimensional euclidean space. We develop a systematic approach to the subject by using the theory of compact Riemann surfaces. Although the results we can prove now are mainly local and analytic in nature, it seems that a global and algebraic theory is possible. One of the important parts of our approach involves the Jacobi inversion problem which leads us to think that our subject is closely related to the special divisor theory of complex algebraic curves. So far, we are able to establish lower bounds for the dimension of certain moduli spaces of minimal immersions under the assumption of the non-emptiness of the moduli space.