Stable constant mean curvature surfaces minimize area
Author : Karsten Grosse-Brauckmann
Publisher :
Page : 8 pages
File Size : 46,90 MB
Release : 1995
Category :
ISBN :
Author : Karsten Grosse-Brauckmann
Publisher :
Page : 8 pages
File Size : 46,90 MB
Release : 1995
Category :
ISBN :
Author : Katsuei Kenmotsu
Publisher : American Mathematical Soc.
Page : 156 pages
File Size : 50,10 MB
Release : 2003
Category : Mathematics
ISBN : 9780821834794
The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space. A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. There is a rich and well-known theory of minimal surfaces. A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses. An easy example of a surface of constant mean curvature is the sphere. A nontrivial example is provided by the constant curvature torus, whose discovery in 1984 gave a powerful incentive for studying such surfaces. Later, many examples of constant mean curvature surfaces were discovered using various methods of analysis, differential geometry, and differential equations. It is now becoming clear that there is a rich theory of surfaces of constant mean curvature. In this book, the author presents numerous examples of constant mean curvature surfaces and techniques for studying them. Many finely rendered figures illustrate the results and allow the reader to visualize and better understand these beautiful objects. The book is suitable for advanced undergraduates, graduate students and research mathematicians interested in analysis and differential geometry.
Author : Bennett Palmer
Publisher :
Page : 156 pages
File Size : 21,49 MB
Release : 1986
Category :
ISBN :
Author : Leung-Fu Cheung
Publisher :
Page : 88 pages
File Size : 29,94 MB
Release : 1991
Category : Geometry, Differential
ISBN :
Author : Rafael López
Publisher : Springer Science & Business Media
Page : 296 pages
File Size : 47,80 MB
Release : 2013-08-31
Category : Mathematics
ISBN : 3642396267
The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary phenomena. Further, as most techniques used in the theory of CMC surfaces not only involve geometric methods but also PDE and complex analysis, the theory is also of great interest for many other mathematical fields. While minimal surfaces and CMC surfaces in general have already been treated in the literature, the present work is the first to present a comprehensive study of “compact surfaces with boundaries,” narrowing its focus to a geometric view. Basic issues include the discussion whether the symmetries of the curve inherit to the surface; the possible values of the mean curvature, area and volume; stability; the circular boundary case and the existence of the Plateau problem in the non-parametric case. The exposition provides an outlook on recent research but also a set of techniques that allows the results to be expanded to other ambient spaces. Throughout the text, numerous illustrations clarify the results and their proofs. The book is intended for graduate students and researchers in the field of differential geometry and especially theory of surfaces, including geometric analysis and geometric PDEs. It guides readers up to the state-of-the-art of the theory and introduces them to interesting open problems.
Author : Steffen Fröhlich
Publisher :
Page : 7 pages
File Size : 32,2 MB
Release : 2002
Category :
ISBN :
Author : Nahid Sultana
Publisher : LAP Lambert Academic Publishing
Page : 112 pages
File Size : 34,43 MB
Release : 2014-02
Category :
ISBN : 9783848481538
The field of Constant Mean Curvature (CMC) surfaces had its beginning in the nineteenth century with the works of Riemann, Weierstrass and Enneper. Recently it has enjoyed a surge of growth due to the advent of computer graphics. This field has applications in many applied fields such as applied physics, polymer science, architecture, and computer graphics. The method for the construction of CMC surfaces was developed by J. Dorfmeister, F. Pedit, and H. Wu; it is commonly called the DPW method. The DPW method is a Weierstrass type representation for CMC surfaces, using techniques of integrable systems. It gives an algorithm to compute all CMC surfaces. This book includes: explicit conformal parametrizations of CMC surfaces of revolution, in each of the three space forms Euclidean 3-space, spherical 3-space and hyperbolic 3-space by using the DPW method; the lower bounds for the Morse index and nullity of CMC tori of revolution in the 3-sphere; the spectra of Jacobi operators for CMC tori of revolution in the 3-sphere; stability properties of CMC surfaces of revolution in general simply-connected spherically symmetric 3-spaces, and in the particular case of Schwarzschild space.
Author : Robert Osserman
Publisher : Courier Corporation
Page : 226 pages
File Size : 48,73 MB
Release : 2013-12-10
Category : Mathematics
ISBN : 0486167690
Newly updated accessible study covers parametric and non-parametric surfaces, isothermal parameters, Bernstein’s theorem, much more, including such recent developments as new work on Plateau’s problem and on isoperimetric inequalities. Clear, comprehensive examination provides profound insights into crucial area of pure mathematics. 1986 edition. Index.
Author : Steffen Fröhlich
Publisher :
Page : 7 pages
File Size : 37,79 MB
Release : 2002
Category :
ISBN :
Author : Ben Chow
Publisher : CRC Press
Page : 216 pages
File Size : 30,10 MB
Release : 1996-10-15
Category : Mathematics
ISBN : 1439864519
This book documents the results of a workshop held at the Geometry Center (University of Minnesota, Minneapolis) and captures the excitement of the week.