Convex hull properties of harmonic maps
Convex hull properties of harmonic maps
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DOI：
10.4310/jdg/1214460861
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发表时间：
1998
影响因子：
2.5
通讯作者：
Peter Li;Jiaping Wang
中科院分区：
文献类型：
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作者：
Peter Li;Jiaping Wang
1975年，Yau [Y]通过梯度估计证明了具有非负Ricci曲率的完备流形M必须满足调和函数的强Liouville性质。强刘维尔性质（Liouville property）断言定义在M上的任何正（有界）调和函数必须是恒常的。1980年，Cheng [C]将梯度估计推广到从具有非负Ricci曲率的流形M到Cartan-Hadamard流形N的调和映射。特别是，调和映射的刘维尔性质可以推导出这种情况。调和映射的刘维尔性质断言，如果调和映射的像包含在有界集中，则该映射必须恒同。事实上，Cheng的梯度估计实际上产生了一个稍微更强的定理。这意味着，如果从具有非负Ricci曲率的流形到Cartan-Hadamard流形的调和映射是次线性增长的，则该映射一定是常数。地图u：M → N是次线性增长的，如果存在一个点p ∈ M和一个点o ∈ N，使得u的象到点o的距离d（u（x），o）满足d（u（x），o）= o（ρ（x）），
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)),