Memoirs of the American Mathematical Society 2012; 105 pp; softcover Volume: 218 ISBN10: 0821853570 ISBN13: 9780821853573 List Price: US$70 Individual Members: US$42 Institutional Members: US$56 Order Code: MEMO/218/1023
 The central theme of this paper is the variational analysis of homeomorphisms \(h: {\mathbb X} \overset{\text{onto}}{\longrightarrow} {\mathbb Y}\) between two given domains \({\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n\). The authors look for the extremal mappings in the Sobolev space \({\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})\) which minimize the energy integral \[{\mathscr E}_h=\int_{{\mathbb X}} \ Dh(x) \^n\, \mathrm{d}x\,.\] Because of the natural connections with quasiconformal mappings this \(n\)harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal \(n\)harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art. Table of Contents  Introduction and overview
Part 1. Principal Radial \(n\)Harmonics  Nonexistence of \(n\)Harmonic homeomorphisms
 Generalized \(n\)Harmonic mappings
 Notation
 Radial \(n\)harmonics
 Vector calculus on annuli
 Free Lagrangians
 Some estimates of free Lagrangians
 Proof of Theorem 1.15
Part 2. The \(n\)Harmonic Energy  Harmonic mappings between planar annuli, Proof of Theorem 1.8
 Contracting Pair, \(\mbox{Mod}\, {\mathbb A}^{\! \ast} \leqslant \mbox{Mod}\, {\mathbb A}\)
 Expanding Pair, \(\mbox{Mod}\, {\mathbb A}^{\! \ast} > \mbox{Mod}\, {\mathbb A}\)
 The uniqueness
 Above the upper Nitsche bound, \(n \geqslant 4\)
 Quasiconformal mappings between annuli
 Bibliography
