HARMONIC-FUNCTIONS ON COMPLETE RIEMANNIAN MANIFOLDS
HARMONIC-FUNCTIONS ON COMPLETE RIEMANNIAN MANIFOLDS
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DOI：
10.1002/cpa.3160280203
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发表时间：
1975-01-01
影响因子：
3
通讯作者：
YAU, ST
中科院分区：
文献类型：
--
作者：
YAU, ST
众所周知，人们可以将开黎曼曲面分为两类：允许非常数有界调和函数的开黎曼曲面和不允许非常数有界调和函数的开黎曼曲面。事实证明，这种分类也可以描述在一定的完整的黎曼度量的曲率，导致共形结构在这个表面上。事实上，Blanc-Fiala-Huber [2]的一个定理指出，具有非负曲率的完备二维黎曼流形不允许有非常数有界调和函数。另一方面，Ahlfors的一个定理指出，一个完全的单连通二维黎曼流形，其曲率从上到下由一个负常数限定，不允许一个非常数有界调和函数。在本文中，我们将研究高维黎曼流形上的类似问题。我们注意到，如果我们在欧氏空间中的一个区域上配备一定的黎曼度量，我们在这里所做的相当于研究这个区域上的某些椭圆方程的行为。如果我们试图将黎曼曲面上的理论推广到高维空间，我们遇到的第一个困难是我们不太了解拓扑结构，而且我们通常没有复杂的结构。即使我们有复杂的结构，也没有一致化定理让我们能够使用函数论。
It is well known that one can classify open Riemann surfaces into two classes: those which admit a non-constant bounded harmonic function and those which do not admit such a function. It turns out that this classification can also be described in terms of curvature of a certain complete Riemannian metric which induces the conformal structure on this surface. In fact, a theorem of Blanc-Fiala-Huber [2] says that a complete two-dimensional Riemannian manifold with non-negative curvature does not admit a non-constant bounded harmonic function. On the other hand, a theorem of Ahlfors states that a complete simply connected two-dimensional Riemannian manifold with curvature bounded from above by a negative constant does admit a non-constant bounded harmonic function. In this paper, we shall study similar problems on a higher-dimensional Riemannian manifold. We remark that, if we equip a domain in Euclidean space with a certain Riemannian metric, what we are doing here amounts to studying the behavior of certain elliptic equations on this domain. If we try to generalize the theory on a Riemann surface to higherdimensional space, the first difficulty we encounter is that we do not know the topology well and we do not have complex structure in general. Even if we have complex structure, there is no uniformization theorem for us to be able to use function theory.