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Theorems on regularity and singularity of energy minimizing maps. Based on lecture notes by Norbert Hungerbühler. (English) Zbl 0864.58015
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. vii, 152 p. (1996).
The present book gives a comprehensive introduction into some recent developments of the regularity theory for harmonic maps between Riemannian manifolds. In sections one and two the author first collects some auxiliary material concerning function spaces and linear elliptic theory which is used to give a new proof of the so-called \(\varepsilon\)-regularity theorem for energy minimizing maps. This result was presented first in [J. Differ. Geom. 17, 307-335 (1982; Zbl 0521.58021)] by R. Schoen and K. Uhlenbeck. The approach of the author is very clear and understandable even for non-experts in the field of harmonic maps.
The second part of the book is now devoted to the study of the singular set of minimizing maps including a discussion of tangent maps and the rectifiability properties of the singular set. For example, the following result is demonstrated: if \(u\) is an energy minimizing map \(\Omega\to N\) from some domain \(\Omega\) in \(\mathbb{R}^n\) into a real-analytic manifold \(N\), then for each closed ball \(B\subset \Omega\), the intersection of \(B\) with the singular set of \(u\) is the union of a finite pairwise disjoint collection of locally \((n-3)\)-rectifiable locally compact subsets.

58E20 Harmonic maps, etc.
49N60 Regularity of solutions in optimal control
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory