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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.

##### MSC:
 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