Uniqueness of the Functional Determinant
Uniqueness of the Functional Determinant
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DOI：
10.1007/s002200050223
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发表时间：
1997-11
影响因子：
2.4
通讯作者：
M. Gursky
中科院分区：
文献类型：
--
作者：
M. Gursky
共形拉普拉斯算子的函数行列式和狄拉克算符的平方在所有共形度规中的四个球面的标准圆度规上极值(直到度量等价)。在本文中，我们证明了这是唯一的临界点，从而推广了Onofri和Osgood，Phillips和Sarnak关于S2上的泛函行列式的工作，该工作刻画了常曲率度量是行列式的唯一临界点。此外，我们还引入了一个新的对称双张量场，它定义在任何共形平坦的四维流形上，可以看作是爱因斯坦引力张量的四阶推广。作为结果，我们证明了具有边界的流形上的Pohozaev恒等式，其中流形允许共形Killing场。
The functional determinant of the conformal laplacian and the square of the Dirac operator are known to be extremized at the standard round metric of the four-sphere among all conformal metrics (up to gauge equivalence). In this article we show that this is the unique critical point, thus extending the work of Onofri and Osgood, Phillips and Sarnak for the functional determinant onS2which characterized the constant curvature metric as the unique critical point of the determinant. In addition, we introduce a new symmetric two-tensor field which is defined on any conformally flat four-manifold and can be viewed as a fourth order generalization of the Einstein gravitational tensor. As a consequence we prove a Pohozaev identity for manifolds with boundary which admit conformal Killing vector fields.