R^3中的完備極小子流形

Abstract

在這篇論文裡一共有七節，在第一節中，介紹了什麼是極小曲面，在第二節裡，給出了製造極小曲面的有力工具，進而在進入第三節時，造出了一些極小曲面的具體例子並說明何謂完備的極小曲面，到了第四節，給予了完備極小曲面的 Guass map 其性質的討論，進入第五節，首先，透過 Osserman 的定理，我們將焦點集中在具有有限全曲率的完備極小曲面上，接著，將目光礎b end 的行為表現上，來到了第六節，對具有有限全曲率的完備極小曲面做了點分類及給了些曲面不存在的情況，到了最後一節，造出了較第三節要多樣的例子，並在最後列表做了例子間的比較。

There are seven sections in this paper. In the first one, we introduce what we mean by a minimal surface. In the second, a powerful tool for constructing minimal surfaces is given, and then, in the third, some minimal surfaces are constructed from it and the definition of a complete minimal surface is given, too. In the fourth, the properties of the Gauss maps of complete minimal surfaces are discussed. In the fifth, by Osserman's theorem, we focus our attation on complete minimal surfaces with finite total curvature at first, and then pay attation to the characters of ends. In the sixth, there are some classifications of complete minimal surfaces with finite total curvature and some conditions which imply the nonexistence of complete minimal surfaces. In the last one, a more variety of surfaces than section 3 are given, and finally we list surfaces in tables to make some comparisons.