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 Functions
- Properties of Dir-Minimizing Q Valued Functions and Tangent Cone Stratification of Mass Minimizing Rectifiable Currents
- Approximation 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 Minimizing
- Approximation 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 Manifold
- Bounds on the Frequency Functions and the Main Interior Regularity Theorem
Readership: Students and researchers dealing with the calculus of variations.
