Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space
Published on 1. January 1991
Book
142 pages
978-0-8218-4552-3 (ISBN)
Description
This book grew out of lectures presented to students of mathematics, physics, and mechanics by A T Fomenko at Moscow University, under the auspices of the Moscow Mathematical Society. The book describes modern and visual aspects of the theory of minimal, two-dimensional surfaces in three-dimensional space. The main topics covered are: topological properties of minimal surfaces, stable and unstable minimal films, classical examples, the Morse-Smale index of minimal two-surfaces in Euclidean space, and minimal films in Lobachevskian space. Requiring only a standard first-year calculus and elementary notions of geometry, this book brings the reader rapidly into this fascinating branch of modern geometry.
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Content
Physical prerequisites: Interfaces between two media; The principle of economy in nature; Classical minimal surfaces in R3: Catenoids; The helicoid; The minimal surface equation, Bernstein's problem, The Scherk surface; Periodic minimal surfaces; Complete minimal surfaces; General properties of minimal surfaces in R3: Isothermal coordinates; Harmonicity and conformality; The Gaussian mapping and the Weierstrass representation; The global Weierstrass representation; Total curvature and complete minimal surfaces; The geometry of complete minimal surfaces of finite total curvature; Indices of two-dimensional minimal surfaces in R3.