Harmonic maps between Riemannian polyhedra. With a preface by M. Gromov. (English) Zbl 0979.31001

Cambridge Tracts in Mathematics. 142. Cambridge: Cambridge University Press. xii, 296 p. (2001).
Given two spaces \(X\) and \(Y\), suppose a harmonic structure on \(X\), that is, a distinguished space of real-valued functions on \(X\) regarded as “harmonic”. Further, one may define another space of “subharmonic functions” on \(X\). In parallel, the functions on \(Y\) are distinguished as “convex functions”. A map \(f:X\to Y\) is called “harmonic” if the pull-back of every “convex function” on \(Y\) is “subharmonic” on \(X\).
The authors specialize the “measurable Riemannian” structures of polyhedra for both \(X\) and \(Y\). The book contains an introduction and three parts: (I) Domains, targets and examples; (II) Potential theory on polyhedra; (III) Maps between polyhedra.


31-02 Research exposition (monographs, survey articles) pertaining to potential theory
31C20 Discrete potential theory
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
52B70 Polyhedral manifolds