## Harmonic Maps Between Riemannian PolyhedraHarmonic maps between smooth Riemannian manifolds play a ubiquitous role in differential geometry. Examples include geodesics viewed as maps, minimal surfaces, holomorphic maps and Abelian integrals viewed as maps to a circle. The theory of such maps has been extensively developed over the last 40 years, and has significant applications throughout mathematics. This 2001 book extends that theory in full detail to harmonic maps between broad classes of singular Riemannian polyhedra, with many examples being given. The analytical foundation is based on existence and regularity results which use the potential theory of Riemannian polyhedral domains viewed as Brelot harmonic spaces and geodesic space targets in the sense of Alexandrov and Busemann. The work sets out much material on harmonic maps between singular spaces and will hence serve as a concise source for all researchers working in related fields. |

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### Contents

Harmonic spaces Dirichlet spaces and geodesic spaces | 15 |

Examples of domains and targets | 30 |

Riemannian polyhedra | 41 |

The Sobolev space W12X | 63 |

Weakly harmonic and weakly subsuperharmonic functions | 72 |

Harnack inequality and Holder continuity for weakly | 79 |

Potential theory on Riemannian polyhedra | 99 |

Examples of Riemannian polyhedra and related spaces | 130 |

Holder continuity of energy minimizers | 178 |

Existence of energy minimizers | 198 |

Harmonic maps Totally geodesic maps | 217 |

Harmonic i nor phis iris | 247 |

Energy according to KorevaarSchoen | 259 |

277 | |

291 | |

Energy of maps | 151 |

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admissible Riemannian polyhedron assume ball bi-Lip boundary Chapter complete geodesic space components continuous map converges convex functions Corollary defined Definition 9.2 denote Dirichlet space embedded energy density equivalent Euclidean Riemannian Euclidean space Example finite energy follows geodesic segment given harmonic functions harmonic map harmonic morphism hence Holder continuous holomorphic homeomorphism homotopy intrinsic distance isometry Lemma linear Lip homeomorphism Lipc(X Lipschitz continuous Lipschitz manifold locally compact metric g metric space minimizing n-simplexes neighbourhood nonpositive curvature null set open set Poincare inequality polar polyhedra proof of Theorem Proposition quasicontinuous quasiopen sets Remark replaced resp Riemann Riemannian manifold Riemannian metric Riemannian structure sequence set U C X simplexwise smooth simplicial simply connected smooth Riemannian manifold Sobolev st(a Subexample subharmonic functions superharmonic function Suppose Theorem 9.1 topology triangulation unique vertex weakly harmonic weakly harmonic function weakly subharmonic Wlo'c Y,dy