Complete embedded minimal surfaces of finite total curvature
Author | : David Hoffman |
Publisher | : |
Total Pages | : 89 |
Release | : 1995 |
Genre | : |
ISBN | : |
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Author | : David Hoffman |
Publisher | : |
Total Pages | : 89 |
Release | : 1995 |
Genre | : |
ISBN | : |
Author | : Kichoon Yang |
Publisher | : Springer Science & Business Media |
Total Pages | : 167 |
Release | : 2013-03-09 |
Genre | : Mathematics |
ISBN | : 9401711046 |
This monograph contains an exposition of the theory of minimal surfaces in Euclidean space, with an emphasis on complete minimal surfaces of finite total curvature. Our exposition is based upon the philosophy that the study of finite total curvature complete minimal surfaces in R3, in large measure, coincides with the study of meromorphic functions and linear series on compact Riemann sur faces. This philosophy is first indicated in the fundamental theorem of Chern and Osserman: A complete minimal surface M immersed in R3 is of finite total curvature if and only if M with its induced conformal structure is conformally equivalent to a compact Riemann surface Mg punctured at a finite set E of points and the tangential Gauss map extends to a holomorphic map Mg _ P2. Thus a finite total curvature complete minimal surface in R3 gives rise to a plane algebraic curve. Let Mg denote a fixed but otherwise arbitrary compact Riemann surface of genus g. A positive integer r is called a puncture number for Mg if Mg can be conformally immersed into R3 as a complete finite total curvature minimal surface with exactly r punctures; the set of all puncture numbers for Mg is denoted by P (M ). For example, Jorge and Meeks [JM] showed, by constructing an example g for each r, that every positive integer r is a puncture number for the Riemann surface pl.
Author | : Kichoon Yang |
Publisher | : Springer |
Total Pages | : 0 |
Release | : 2010-12-05 |
Genre | : Mathematics |
ISBN | : 9789048144433 |
This monograph contains an exposition of the theory of minimal surfaces in Euclidean space, with an emphasis on complete minimal surfaces of finite total curvature. Our exposition is based upon the philosophy that the study of finite total curvature complete minimal surfaces in R3, in large measure, coincides with the study of meromorphic functions and linear series on compact Riemann sur faces. This philosophy is first indicated in the fundamental theorem of Chern and Osserman: A complete minimal surface M immersed in R3 is of finite total curvature if and only if M with its induced conformal structure is conformally equivalent to a compact Riemann surface Mg punctured at a finite set E of points and the tangential Gauss map extends to a holomorphic map Mg _ P2. Thus a finite total curvature complete minimal surface in R3 gives rise to a plane algebraic curve. Let Mg denote a fixed but otherwise arbitrary compact Riemann surface of genus g. A positive integer r is called a puncture number for Mg if Mg can be conformally immersed into R3 as a complete finite total curvature minimal surface with exactly r punctures; the set of all puncture numbers for Mg is denoted by P (M ). For example, Jorge and Meeks [JM] showed, by constructing an example g for each r, that every positive integer r is a puncture number for the Riemann surface pl.
Author | : Yi Fang |
Publisher | : |
Total Pages | : 154 |
Release | : 1990 |
Genre | : Gauss maps |
ISBN | : |
Author | : Robert Osserman |
Publisher | : Springer Science & Business Media |
Total Pages | : 279 |
Release | : 2013-03-14 |
Genre | : Mathematics |
ISBN | : 3662034840 |
Few people outside of mathematics are aware of the varieties of mathemat ical experience - the degree to which different mathematical subjects have different and distinctive flavors, often attractive to some mathematicians and repellant to others. The particular flavor of the subject of minimal surfaces seems to lie in a combination of the concreteness of the objects being studied, their origin and relation to the physical world, and the way they lie at the intersection of so many different parts of mathematics. In the past fifteen years a new component has been added: the availability of computer graphics to provide illustrations that are both mathematically instructive and esthetically pleas ing. During the course of the twentieth century, two major thrusts have played a seminal role in the evolution of minimal surface theory. The first is the work on the Plateau Problem, whose initial phase culminated in the solution for which Jesse Douglas was awarded one of the first two Fields Medals in 1936. (The other Fields Medal that year went to Lars V. Ahlfors for his contributions to complex analysis, including his important new insights in Nevanlinna Theory.) The second was the innovative approach to partial differential equations by Serge Bernstein, which led to the celebrated Bernstein's Theorem, stating that the only solution to the minimal surface equation over the whole plane is the trivial solution: a linear function.
Author | : William Meeks |
Publisher | : Springer Science & Business Media |
Total Pages | : 136 |
Release | : 2002-03-25 |
Genre | : Education |
ISBN | : 9783540431206 |
In the second half of the twentieth century the global theory of minimal surface in flat space had an unexpected and rapid blossoming. Some of the classical problems were solved and new classes of minimal surfaces found. Minimal surfaces are now studied from several different viewpoints using methods and techniques from analysis (real and complex), topology and geometry. In this lecture course, Meeks, Ros and Rosenberg, three of the main architects of the modern edifice, present some of the more recent methods and developments of the theory. The topics include moduli, asymptotic geometry and surfaces of constant mean curvature in the hyperbolic space.
Author | : William Meeks |
Publisher | : American Mathematical Soc. |
Total Pages | : 195 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 0821869124 |
Meeks and Pérez extend their 2011 survey article "The classical theory of Minimal surfaces" in the Bulletin of the American Mathematical Society to include other recent research results. Their topics include minimal surfaces with finite topology and more than one end, limits of embedded minimal surfaces without local area or curvature bounds, conformal structure of minimal surfaces, embedded minimal surfaces of finite genus, topological aspects of minimal surfaces, and Calabi-Yau problems. There is no index. Annotation ©2013 Book News, Inc., Portland, OR (booknews.com).
Author | : Yi Fang |
Publisher | : |
Total Pages | : 192 |
Release | : 1996 |
Genre | : Geometry, Differential |
ISBN | : |
Author | : W.H. III Meeks |
Publisher | : Springer |
Total Pages | : 126 |
Release | : 2004-10-11 |
Genre | : Mathematics |
ISBN | : 3540456090 |
In the second half of the twentieth century the global theory of minimal surface in flat space had an unexpected and rapid blossoming. Some of the classical problems were solved and new classes of minimal surfaces found. Minimal surfaces are now studied from several different viewpoints using methods and techniques from analysis (real and complex), topology and geometry. In this lecture course, Meeks, Ros and Rosenberg, three of the main architects of the modern edifice, present some of the more recent methods and developments of the theory. The topics include moduli, asymptotic geometry and surfaces of constant mean curvature in the hyperbolic space.
Author | : J.Lucas M. Barbosa |
Publisher | : Springer |
Total Pages | : 133 |
Release | : 2006-11-14 |
Genre | : Mathematics |
ISBN | : 3540398309 |