Conformally flat manifolds, Kleinian groups and scalar curvature
Conformally flat manifolds, Kleinian groups and scalar curvature
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DOI：
10.1007/bf01393992
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发表时间：
1988-02
影响因子：
3.1
通讯作者：
R. Schoen;S. Yau
中科院分区：
文献类型：
--
作者：
R. Schoen;S. Yau
二维曲面上的共形结构的自然推广之一是n流形上的共形平面结构。在高维中，并不是所有流形都有这样的结构，因此对共形平面流形进行分类是一个困难的问题。回想一下，共形平坦黎曼流形是其度量与欧几里德度量局部共形等价的流形。Kuiper [Kul， Ku2]是第一个研究这些流形的全局性质的人。用阿贝尔基本群对这些紧共形平面流形进行了分类。对于大于2维，Liouville定理告诉我们S ‘ ’的保形变换是局部确定的，由M6bius变换给出。因此，通过标准单形论证，一个单连通共形平面流形（维数为bbbb3）在与S的M6bius变换复合时具有保形浸入S的唯一性。对于一个一般的局部共形平面流形，我们可以从它的泛盖上确定这样一个共形浸入，这种浸入称为展开映射。流形的基群作用于它的全称盖上，并根据上面的唯一性陈述，通过一个称为完整表示的同态映射到m6群上。发展映射和rc1在MSbius群上的同态是研究保形形流形的最重要的不变量。一类重要的局部共形形流形产生于这样一种情况：展开映射是内射的，因此流形是某些Kleinian群的S n的开子集的商。这类共形fiat流形已被许多数学家广泛研究，包括Mostow， Thurston, Kulkarni, Goldman， Kamishima等人。本文的主要成果之一是我们发现了一类广泛的局部共形形流形，它们的发展映射是内射的。特别地，这种流形是由Kleinian群构成的S '中的单连通域的商。为了解释这类流形，我们需要解释
One of the natural generalizations of conformal structure on a two dimensional surface is a conformally flat structure on an n-manifold. In higher dimensions, not every manifold admits such a structure and it is a difficult problem to give a good classification of conformally flat manifolds. Recall that conformally flat Riemannian manifolds are manifolds whose metrics are locally conformally equivalent to the Euclidean metric. Kuiper [Kul, Ku2] was the first to study the global properties of these manifolds. He classified those compact conformally flat manifolds with abelian fundamental group. For dimensions greater than two, the Liouville theorem tells us that the conformal transformations of S" are determined locally and are given by M6bius transformations. Hence by a standard monodromy argument, a simply connected conformally flat manifold (with dimension> 3) has a conformal immersion into S" which is unique up to composition with a M6bius transformation of S". For a general locally conforreally flat manifold, we can determine such a conformal immersion from its universal cover and this immersion is called the developing map. The fundamental group of the manifold acts on its universal cover and by the above uniqueness statement, is mapped into the M6bius group by a homomorphism called the holonomy representation.The developing map and the homomorphism of rc 1 into the MSbius group form the most important invariants for the study of conformally fiat manifolds. An important class of locally conformally fiat manifolds arises from the case in which the developing maps are injective and the manifolds are therefore quotients of open subsets of S n by certain Kleinian groups. This class of conforrnaUy fiat manifolds has been extensively studied by many mathematicians including Mostow, Thurston, Kulkarni, Goldman, Kamishima and others. One of the main accomplishments of this paper is that we find an extensive class of locally conformally fiat manifolds whose developing maps are injective. Such manifolds are, in particular, quotients of a simply connected domain in S" by a Kleinian group. To explain this class of manifolds, we need to explain