Lanterna Rally

Author: Vladimir Scheffer

Publisher: World Scientific

Published: 2000-06-30

Total Pages: 972

ISBN-13: 9814494119

Fred Almgren exploited the excess method for proving regularity theorems in the calculus of variations. His techniques yielded Hölder continuous differentiability except for a small closed singular set. In the sixties and seventies Almgren refined and generalized his methods. Between 1974 and 1984 he wrote a 1,700-page proof that was his most ambitious development of his ground-breaking ideas. Originally, this monograph was available only as a three-volume work of limited circulation. The entire text is faithfully reproduced here. This book gives a complete proof of the interior regularity of an area-minimizing rectifiable current up to Hausdorff codimension 2. The argument uses the theory of Q-valued functions, which is developed in detail. For example, this work shows how first variation estimates from squash and squeeze deformations yield a monotonicity theorem for the normalized frequency of oscillation of a Q-valued function that minimizes a generalized Dirichlet integral. The principal features of the book include an extension theorem analogous to Kirszbraun's theorem and theorems on the approximation in mass of nearly flat mass-minimizing rectifiable currents by graphs and images of Lipschitz Q-valued functions. Contents:Basic Properties of Q and Q Valued FunctionsProperties of Dir-Minimizing Q Valued Functions and Tangent Cone Stratification of Mass Minimizing Rectifiable CurrentsApproximation in Mass of Nearly Flat Rectifiable Currents which are Mass Minimizing in Manifolds by Graphs of Lipschitz Q Valued Functions Which Can Be Weakly Nearly Dir MinimizingApproximation in Mass of a Nearly Flat Rectifiable Current Which Is Mass Minimizing in a Manifold by the Image of a Lipschitz Q(Rm+n) Valued Function Defined on a Center ManifoldBounds on the Frequency Functions and the Main Interior Regularity Theorem Readership: Students and researchers dealing with the calculus of variations. Keywords:Regularity;Area-Minimizing Surfaces of Codimension Greater Than One;Multiple-Valued Functions;Currents;Center Manifold;Dirichlet's Integral;Frequency FunctionReviews: “The book closes with a number of appendices which also are of independent interest, and it starts with a beautiful Introduction (16 pages) which contains a 'Summary of the principal themes' by chapters … This work is a monument.” Mathematics Abstracts “Now, thanks to the efforts of editors Jean Taylor and Vladimir Scheffer, Almgren's three-volume, 1700-page typed preprint has been published as a single, attractively typset volume of less than 1000 pages … Perhaps advances in knowledge will eventually make possible a much shorter and more transparent proof of Almgren's theorem. But I suspect that if such a proof is discovered, it will still use the basic approach and many of the tools pioneered by Almgren in this monumental work.” Mathematical Reviews