The Gauss map and total curvature of complete minimal Lagrangian surfaces in the complex two-space

The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagrangian surface with finite total curvature in the complex two-space. Moreover, we prove that if the Gauss map of a complete minimal Lagrangian surface which is not a Lagrangian plane omits three values, then it takes all other values infinitely many times.Comment: 11 page

Research of Impact Manipulation

科学研究費助成事業(科学研究費補助金)研究成果報告書：基盤研究（C）2009-2012課題番号：2156025

Technical Note : Debian GNU/Linux Install

We describe a linux-based operating system (OS): Debian install to a personal computer (PC). We examine all the steps necessary to a console type PC, and a full desktop type PC. You can modify a console type to various server, like file, web, dns, and etc. server. A desktop can be a minimum configuration to enable to add software as we go by, or a fully configured desktop with some popular software like office suites, web browser, file viewer, and etc. The benefit of successful installation is free, and fully supported modern OS installed to reasonably old, but discarded PCs due to the terminated Windows OS support. Installing OS to any PC can be a puzzle to be solved. It is similar to Operations Research problem solving: Goal is set, but there may exist some restrictions

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space. © 2017 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin HeidelbergEmbargo Period 12 month

Timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space

It is shown that timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in PSL(2,R) via Bryant type representation formulae. These formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson correspondence, between timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space and timelike minimal surfaces in Minkowski 3-space. The hyperbolic Gauss map of timelike surfaces in anti-de Sitter 3-space, which is a close analogue of the classical Gauss map is considered. It is discussed that the hyperbolic Gauss map plays an important role in the study of timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space. In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauss map and timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space is studied.Comment: 47 pages, 24 figures, references revised, Annals of Global Analysis and Geometr