Conformally invariant powers of the Laplacian — A complete nonexistence theorem
Conformally invariant powers of the Laplacian — A complete nonexistence theorem
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DOI：
10.1090/s0894-0347-04-00450-3
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发表时间：
2004-01
影响因子：
3.9
通讯作者：
A. Gover;K. Hirachi
中科院分区：
文献类型：
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作者：
A. Gover;K. Hirachi
共形不变算子及其确定的方程在研究具有伪黎曼、黎曼、共形及相关结构的流形中起着核心作用。这一发现至少可以追溯到上个世纪的早期，当时人们发现，弯曲时空中无质量粒子的方程表现出共形不变性。在这种情况下，一个关键算子是具有一个伪拉普拉斯算子的前项的共形不变波算子。该算子的黎曼签名变体是紧流形上Yam abe问题的一个基本工具。在这里，我们试图找到一个度规，从一个给定的共形类中，它具有恒定的标量曲率。最近，这些算子的高阶类似物，即带拉普拉斯次幂的加权函数（即共形密度）上的共形不变算子，在生成和解决其他曲率处方问题以及几何谱理论和数学物理中的其他问题中起着核心作用[2,5,15]。在平面环境下，这些算子的存在可以追溯到[16]，其中表明，在四维闵可夫斯基空间上，对于k G N ={1,2，…}，平波算符Ak的k次幂，作用于适当权的密度，在保形群的作用下是不变的。更一般地说，如果？[w]表示权值为1的保形密度空间，则在维数为n > 3的平坦保形流形（任意签名）上，对于每一个k E n，存在一个唯一的保形不变算子
Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observation dates back to at least the very early part of the last century when it was shown that the equations of massless particles on curved space-time exhibit conformai invariance. In this setting a key operator is the con formally invariant wave operator which has leading term a pseudo-Laplacian. The Riemannian signature variant of this operator is a fundamental tool in the Yam abe problem on compact manifolds. Here one seeks to find a metric, from a given conformai class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformai densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2, 5, 15]. In the flat setting, the existence of such operators dates back to [16], where it is shown that, on 4-dimensional Minkowski space, for k G N = {1,2,...}, the kth power of the flat wave operator Ak, acting on densities of the appropriate weight, is invariant under the action of the conformai group. More generally, if ?[w] denotes the space of conformai densities of weight uiGl, then on a flat conformai manifold of dimension n > 3 (and any signature) there exists, for each k E N, a unique conformally invariant operator