This paper is a study of harmonic maps from Riemannian polyhedra to locally non-positively curved geodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functions and harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau Theorem on a complete pseudomanifolds with non-negative Ricci curvature. Then we study 2-parabolic admissible Riemannian polyhedra and prove some vanishing results on them.
Abstract. We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, M j , which converge to a nonzero integral current space, M ∞ , in the intrinsic flat sense. We provide examples demonstrating that the covering spaces and covering spectra need not converge in this setting. In fact we provide a sequence of simply connected M j diffeomorphic to S 4 that converge in the intrinsic flat sense to a torus S 1 × S 3 . Nevertheless, we prove that if the δ-covers, M
Abstract. In this paper we prove a compactness theorem for sequences of harmonic maps which are defined on converging sequences of Riemannian manifolds.Harmonic maps are critical points of the energy functional defined on the space of maps between Riemannian manifolds. This theory was developed by J. In this paper, we are going to study the behavior of harmonic maps under convergence. Let M(n, D) denote the set of all compact Riemannian manifolds (M, g) such that dim(M ) = n, diam(M ) < D, and the sectional curvature sec g satisfies | sec g | ≤ 1, equipped with the measured Gromov-Hausdorff topology. Let (M i , g i , dvol M i ) in M(n, D) be a sequence of manifolds which converges to a smooth metric measure space (M, g, Φ dvol M ).) is a sequence of harmonic maps. We are interested in knowing under what circumstances the f i converge to a harmonic map f on the smooth metric measure space (M, g, Φ dvol M ).When a sequence of manifolds (M i , g i ) in M(n, D) converges to a metric space X, according to Fukaya [Fuk88], X is a quotient space Y /O(n), where Y is a smooth manifold. Indeed Y is the limit point of the sequence of frame bundles, F (M i ), over the manifolds M i and X has the structure of a Riemannian polyhedron (X, g X , Φ X µ g ) where µ g is the Riemannian volume element related to the metric g X on X.We state the main result of this paper which is a compactness theorem for sequences of harmonic maps.Theorem 0.1. . Let (M i , g i ) be a sequence of smooth Riemannian manifolds in M(n, D) which converges to a metric measure space (X, g, Φµ g ) in the measured Gromov-Hausdorff
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