Curvature Functions for Open 2-Manifolds
Curvature Functions for Open 2-Manifolds
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DOI：
10.2307/1970898
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发表时间：
1974-03
影响因子：
4.9
通讯作者：
J. Kazdan;F. W. Warner
中科院分区：
文献类型：
--
作者：
J. Kazdan;F. W. Warner
文[12]中提出的基本问题是描述给定的二维流形M可以具有的高斯曲率函数集。本文考虑非紧流形上的这个问题。除了Cohn-Vossen[4]的Gauss-Bonnet型不等式(另见[6]，[8])对于非紧流形上的某些完备度量成立外，非紧2-流形上的高斯曲率函数没有已知的先验限制。事实上，Gromov最近证明了任何非紧2-流形都有严格正曲率的度量和严格负曲率的度量[7]。因此，作为[12]问题1的类比，似乎很自然地会问以下问题：
The basic problem posed in [12] is that of describing the set of Gaussian curvature functions which a given 2-dimensional manifold M can possess. In this paper we consider this problem for the case of non-compact M. Other than the Gauss-Bonnet type inequality of Cohn-Vossen [4] (see also [6], [8]), which holds for certain complete metrics on non-compact manifolds, there is no known a priori restriction for a Gaussian curvature function on a non-compact 2-manifold. Indeed, Gromov has recently shown that any non-compact 2-manifold possesses a metric of strictly positive curvature as well as a metric of strictly negative curvature [7]. As the analogue of Question 1 of [12] it therefore seems natural to ask the following: