论文标题
封闭的四个manifolds上的帕尼尼茨操作员的极端指标
Extremal metrics for the Paneitz Operator on closed four-Manifolds
论文作者
论文摘要
令$（m^4，g）$为尺寸四的封闭式riemannian歧管。我们研究了指标的性质，这是帕尼兹操作员特征值的关键点，当将其视为具有固定体积的Riemannian指标空间的功能。我们证明，上述功能的临界指标仅限于共形类别，这与谐波图（称为外部共形谐波图）的高阶类似物与圆形球体有关。这扩展到封闭式的封闭表面上的四个manifolds众所周知的结果，在共形类别中最大化拉普拉斯特征值，并存在谐波图中的谐波图。还研究了一般关键点（不限于共形类别）的情况，并提供了部分表征。
Let $(M^4,g)$ be a closed Riemannian manifold of dimension four. We investigate the properties of metrics which are critical points of the eigenvalues of the Paneitz operator when considered as functionals on the space of Riemannian metrics with fixed volume. We prove that critical metrics of the aforementioned functional restricted to conformal classes are associated with a higher-order analog of harmonic maps (known as extrinsic conformal-harmonic maps) into round spheres. This extends to four-manifolds well-known results on closed surfaces relating metrics maximizing laplacian eigenvalues in conformal classes with the existence of harmonic maps into spheres. The case of general critical points (not restricted to conformal classes) is also studied, and partial characterization of these is provided.
