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A Theory of Branched Minimal Surfaces (Springer Monographs in Mathematics)by Anthony Tromba
Synopses & ReviewsPublisher Comments:One of the most elementary questions in mathematics is whether an area minimizing surface spanning a contour in three space is immersed or not; i.e. does its derivative have maximal rank everywhere. The purpose of this monograph is to present an elementary proof of this very fundamental and beautiful mathematical result. The exposition follows the original line of attack initiated by Jesse Douglas in his Fields medal work in 1931, namely use Dirichlet's energy as opposed to area. Remarkably, the author shows how to calculate arbitrarily high orders of derivatives of Dirichlet's energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations, where normally only the second derivative or variation is calculated.
Synopsis:This book shows how to calculate arbitrarily high orders of derivatives of the Douglas Energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations.
Table of Contents1.Introduction. 2.Higher order Derivatives of Dirichlets' Energy. 3.Very Special Case; The Theorem for n + 1 Even and m + 1 Odd . 4.The First Main Theorem; NonExceptional Branch Points. 5.The Second Main Theorem: Exceptional Branch Points; The Condition k > l. 6.Exceptional Branch Points Without The Condition k > l. 7.New Brief Proofs of the GulliverOssermanRoyden Theorem . 8.Boundary Branch Points. Scholia. Appendix. Bibliography.
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