Conformal deformation of a Riemannian metric to constant scalar curvature
Conformal deformation of a Riemannian metric to constant scalar curvature
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DOI：
10.4310/jdg/1214439291
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发表时间：
1984
影响因子：
2.5
通讯作者：
R. Schoen
中科院分区：
文献类型：
--
作者：
R. Schoen
微分几何中一个著名的开放问题是给定的紧黎曼流形是否一定等价于常标量曲率的流形。这个问题被称为Yamabe问题，因为它是由Yamabe[8]在1960年提出的，虽然Yamabe的论文声称肯定地解决了这个问题，但1968年N. Trudinger[8]发现Yamabe的论文是严重错误的。在标量曲率为非正的情况下，Trudinger能够修正Yamabe的证明。1976年，T. Aubin b[1]在正标量曲率的情况下取得了进展。Aubin证明了如果dim M >6和M不共形平坦，则M可以共形变为常数标量曲率。到目前为止，Aubin的方法还没有给出关于3、4和5维的Yamabe问题的信息。此外，他的方法只利用了M在一个点的小邻域中的局部几何，因此不能用于共形平面流形，因为Yamabe问题显然是一个全局问题。最近，许多几何学家对正标量曲率的共形平坦流形很感兴趣，其中Yamabe问题的解给出了常数标量曲率的共形平坦度规，这是一个几何上感兴趣的度规。注意到正标量曲率的共形平面流形在连通和的作用下是闭合的，因此包含了具有S X S~副本的球面空间形式的连通和。在本文中，我们引入了一种新的全局思想，并在所有剩余情况下对其进行了肯定解；也就是说，我们断言方程M上存在一个正解u
A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M > 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of S X S~. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation