Harmonic maps between Riemannian polyhedra / J. Eells, B. Fuglede / with a pref. by M. Gromov

Bibliogr. p. [277]-290. Index

The book consists of three parts. After an introductory survey and some analytic and geometric preliminaries, the key notion of a Riemannian polyhedron is introduced in Chapter 4. Essentially, a Riemannian polyhedron is a simplicial complex equipped with an intrinsic metric that comes from a measurable Riemannian structure with locally uniform ellipticity bounds.
The second part of the book is devoted to potential theory on Riemannian polyhedra. Weakly harmonic functions are defined as Sobolev functions minimizing an appropriate Dirichlet energy. Following the classical approach by Moser, the authors then establish a Harnack inequality for weakly harmonic functions (Theorem 6.1). An immediate consequence is the fact that, after a correction on a null set, a weakly harmonic function on a Riemannian polyhedron is Hölder continuous (Theorem 6.2). The authors proceed to define harmonic functions as weakly harmonic functions that are continuous and prove that a Riemannian polyhedron with its harmonic functions forms a harmonic space in the sense of Brelot.
In the third and main part of the book, harmonic maps make their appearance. Here the class of source spaces is restricted to Riemannian polyhedra with piecewise smooth Riemannian structure. The crucial concept is the notion of an energy E(f) for maps between metric spaces (Definition 9.1). A continuous mapping is then called harmonic if it is locally energy minimizing in an appropriate sense (Definition 12.1). In order to prove regularity and existence results one has to impose further restrictions on the target space of the harmonic map, namely that it be non-positively curved. This assumption leads to Hölder continuity of harmonic maps (Theorem 10.1) and a satisfactory existence theory (Theorem 11.1). Finally, the related concepts of totally geodesic maps and harmonic morphisms are discussed. (MathScinet)