| Complete Minimal Surfaces of Finite Total Curvature 1994 Edition Contributor(s): Kichoon Yang (Author) |
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ISBN: 0792330129 ISBN-13: 9780792330127 Publisher: Springer OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: July 1994 Annotation: 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. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Differential - Mathematics | Geometry - Algebraic - Mathematics | Functional Analysis |
| Dewey: 516.362 |
| LCCN: 94027704 |
| Series: Mathematics and Its Applications |
| Physical Information: 0.44" H x 6.14" W x 9.21" (0.92 lbs) 160 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |