Convex Hull Properties of Harmonic Maps
نویسندگان
چکیده
In 1975, Yau [Y] proved, by way of a gradient estimate, that a complete manifold M with non-negative Ricci curvature must satisfy the strong Liouville property for harmonic functions. The strong Liouville property (Liouville property) asserts that any positive (bounded) harmonic function defined on M must be identically constant. In 1980, Cheng [C] generalized the gradient estimate to harmonic maps from a manifold M with non-negative Ricci curvature to a Cartan-Hadamard manifold N . In particular, the Liouville property for harmonic maps can be derived for this situation. The Liouville property for harmonic maps asserts that if the image of the harmonic map is contained in a bounded set, then the map must be identically constant. In fact, Cheng’s gradient estimate actually yields a slightly stronger theorem. It implies that if a harmonic map from a manifold with non-negative Ricci curvature into a Cartan-Hadamard manifold is of sublinear growth then the map must be constant. A map u : M → N is of sublinear growth if there exists a point p ∈ M and a point o ∈ N such that the distance d(u(x), o) between the image of u to the point o satisfies d(u(x), o) = o(ρ(x)),
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