TAILIEUCHUNG - Đề tài " The uniqueness of the helicoid "

In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R3 . M of finite topology means M is homeomorphic to a compact surface M (of genus k and empty boundary) minus a finite number of points p1 , ., pj ∈ M , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends sufficiently small so they are topologically S 1 × [0, 1) and hence, annular. We remark that M is orientable since M is properly. | Annals of Mathematics The uniqueness of the helicoid By William H. Meeks IIIn and Harold Rosenberg Annals of Mathematics 161 2005 727 758 The uniqueness of the helicoid By William H. Meeks III and Harold Rosenberg In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R3. M of finite topology means M is homeomorphic to a compact surface M of genus k and empty boundary minus a finite number of points p1 . pj G M called the punctures. A closed neighborhood E of a puncture in M is called an end of M. We will choose the ends sufficiently small so they are topologically S 1 X 0 1 and hence annular. We remark that M is orientable since M is properly embedded in R3. The simplest examples discovered by Meusnier in 1776 are the helicoid and catenoid and a plane of course . It was only in 1982 that another example was discovered. In his thesis at Impa Celso Costa wrote down the Weierstrass representation of a complete minimal surface modelled on a 3-punctured torus. He observed the three ends of this surface were embedded one top catenoid-type end1 one bottom catenoid-type end and a middle planar-type end2 8 . Subsequently Hoffman and Meeks 15 proved this example is embedded and they constructed for every finite positive genus k embedded examples of genus k and three ends. In 1993 Hoffman Karcher and Wei 14 discovered the Weierstrass data of a complete minimal surface of genus one and one annular end. Computer generated pictures suggested this surface is embedded and the end is asymptotic to an end of a helicoid. Hoffman Weber and Wolf 17 have now given a proof that there is such an embedded surface. Moreover computer evidence suggests that one can add an arbitrary finite number k of handles to a helicoid to obtain a properly embedded genus k minimal surface asymptotic to a helicoid. For many years the search went on for simply connected examples other than the plane and helicoid. We shall prove that there are no such examples.

• Thạc sĩ - Tiến sĩ - Cao học
• helicoid
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center