A Complete Minimal Surface Between Two Parallel Planes In R～3

The theory of complete minimal surfaces in R~3 was a beautiful topic of differential geometry. In 1960's , Calabi asked if it was possible to have a minimal surface entirely contained in a halfspace of R~3 . More discoveries and publications appeared in the past twenties years . In 1980 , the first nonflat complete minimal surface contained in a slab of R~3 was showed by Jorge & Xaxier ([23]). The proof of them ingeniously use Runge's theorem: There exist nonflat complete minimal surfaces contained in a slab of R~3 . This paper is devoted to some respects to a kind of complete minimal surfaces contained in a slab of R~3 .In 1988 , W. H. Meeks III ([33]) has given minimal mobius strip . By Lopez transform , I give a complete nonorientable minimal surfaces contained in a slab of R~3 .especially , when m = 3 , the result was given by F. Ldpez[26]. Secondly , according to the way of F. Brito[5], I take another form of Power series with Hadamard gaps in D. Gnuschke & CH. Pommerenke[20], to give the other form of hyperbolic type minimal surfaces in a slab of R~3 and its precise form similarly.The thesis consists of four parts . The first section briefly introduce the history of minimal surface. In the second section, we give a summary of some basic knowledge and results which can be used for the further study . The third section we only give some relevant results and discussion about bounded, complete , nonflat minimal surfaces in R~3 . Main results is given in the fourth section . Using the way of Lopez ([26])& W. H. Meeks III ([33]) and Brito ([5]), we briefly study a hyperbolic complete minimal surface contained in a slab of R~3 .