Complete Minimal Surfaces of Finite Total Curvature / Edition 1

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This monograph is based on the idea that the study of complete minimal surfaces in R3 of finite total curvature amounts to the study of linear series on algebraic curves. A detailed account of the Puncture Number Problem, which seeks to determine all possible underlying conformal structures for immersed complete minimal surfaces of finite total curvature, is given here for the first time in book form. Several recent results on the puncture number problem are given along with numerous examples. The emphasis is on manufacturing minimal surfaces from a given Riemann surface using the theory of divisions and residue calculus. Relevant results from algebraic geometry are collected in Chapter 1, which makes the book nearly self-contained. A brief survey of minimal surface theory in general is given in Chapter 2. Chapter 3 includes Mo's recent moduli construction.
For graduate students and research mathematicians in differential geometry, function theory and algebraic curves, as well as for those working in materials science or crystallography.

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Editorial Reviews

Based on the idea that the study of the title subject amounts to the study of linear series on algebraic curves. Presents the first book-length treatment of the puncture number problem, which seeks to determine all possible underlying conformational structures for immersed complete minimal surfaces of finite total curvature. Addressed to graduate students and research mathematicians in differential geometry, function theory, algebraic curves, materials science, or crystallography. Annotation c. Book News, Inc., Portland, OR
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Product Details

  • ISBN-13: 9780792330127
  • Publisher: Springer Netherlands
  • Publication date: 7/31/1994
  • Series: Mathematics and Its Applications (closed) Series, #294
  • Edition description: 1994
  • Edition number: 1
  • Pages: 160
  • Product dimensions: 0.50 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface. 1: Background Material. 1.1. Simplicial Homology. 1.2. Complex Algebraic Varieties. 1.3. Compact Riemann Surfaces. 1.4. The Brill-Noether Theorem. 2: Minimal Surfaces: General Theory. 2.1. Intrinsic Surface Theory. 2.2. The Method of Moving Frames. 2.3. The Gauss Map and the Weierstrass Representation. 2.4. The Chern-Osserman Theorem. 2.5. Examples. 2.6. Bernstein Type Theorems. 2.7. Stability of Complete Minimal Surfaces. 3: Minimal Surfaces with Finite Total Curvature. 3.1. The Puncture Number Problem. 3.2. Moduli Space of Algebraic Minimal Surfaces. Bibliography. Index.

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