Conformality of rotationally symmetric maps
Conformality of rotationally symmetric maps
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旋转对称映射的共形性
DOI：
10.1016/j.geomphys.2022.104575
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发表时间：
2022
影响因子：
1.5
通讯作者：
Nobumitsu Nakauchi
中科院分区：
文献类型：
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作者：
佐藤進;佐藤進;Nobumitsu Nakauchi
摘要 设 (M, g),(N, h) 为无边界黎曼流形，f 为从 M 到 N 的平滑映射。我们考虑一个协变对称张量 T f= f⁎ h− 1 m‖ d f‖ 2 g，其中 f⁎ h 表示 f 对度量 h 的拉回，m 是流形 M 的维数。当且仅当映射 f 是弱共角时，张量 T f 才消失。地图。范数“T f”是衡量 f 在每个点的共形性的量。在[4]中，作者介绍了在M紧致的情况下函数E conf (f)=∫ M‖ T f‖ 2 d v g 的临界点的映射。我们称这种地图为 C-平稳地图。在 M 是非紧的情况下，如果 f 在 M 的任何紧子集上都是 C 平稳的，则 f 被定义为 C 平稳映射。任何共形映射或更一般地任何弱共形映射都是 C 平稳映射。然而，存在一个不是弱共形的C-平稳映射。（参见[3]中的例子，第156页。）在源流形或目标流形是n维标准球（n≥5）的情况下，任何稳定的C-平稳映射都是弱共形的。（参见[3]中的定理1和2。）在本文中，我们关注旋转对称映射。我们证明 4 维模型空间之间的任何旋转对称平滑映射当且仅当它是共形映射时才是 C 平稳映射。
Abstract Let (M, g),(N, h) be Riemannian manifolds without boundary, and let f be a smooth map from M into N. We consider a covariant symmetric tensor T f= f⁎ h− 1 m‖ d f‖ 2 g, where f⁎ h denotes the pullback of the metric h by f, and m is the dimension of the manifold M. The tensor T f vanishes if and only if the map f is a weakly conformal map. The norm‖ T f‖ is a quantity which is a measure of the conformality of f at each point. In [4] the author introduced maps which are critical points of the functional E conf (f)=∫ M‖ T f‖ 2 d v g in the case that M is compact. We call such maps C-stationary maps. In the case that M is non-compact, f is defined to be a C-stationary map if it is C-stationary on any compact subset of M. Any conformal map or more generally any weakly conformal one is a C-stationary map. However, there exists a C-stationary map which is not a weakly conformal one.(See Example in [3], p. 156.) In the case that the source manifold or the target one is the n-dimensional standard sphere (n≥ 5), any stable C-stationary map is a weakly conformal one.(See Theorem 1 and 2 in [3].) In this paper we are concerned with rotationally symmetric maps. We prove that any rotationally symmetric smooth map between 4-dimensional model spaces is a C-stationary map if and only if it is a conformal one.