Harmonic Maps and Totally Geodesic Maps Between Metric Spaces |
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Alexandrov space B₁ Banach space Bru(p Cauchy sequence chapter Cheeger-type compact continuous map convex Corollary curvature upper bound d₁ define Definition denote doubling condition dx y1 dx(x dx(y dµ(w dµg EC-continuously harmonic Ep(u exists expp Finslerian function g geodesically complete H¹²(U H¹P(U harmonic maps Hence holds homothetic isometric embedding Lemma lim inf Lip dx Lip u(z Lipschitz continuous Lipu Lº(U locally Lipschitz continuous LP(U map between Riemannian maps between metric measure space metric measure space metric space minimal generalized upper minimal geodesic obtain Poincaré inequality Proof Rademacher's theorem respectively Riemannian manifold RK-domain satisfies the doubling small ɛ Sobolev spaces space of curvature sufficiently small sup 9dx superrigidity Theorem totally geodesic map u₁ unit speed curve upper gradient variation formula xo)² y₁ Zxyz αμ