Weak conformality of stable stationary maps for a functional related to conformality
Weak conformality of stable stationary maps for a functional related to conformality
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与共形性相关的函数的稳定平稳映射的弱共形性
DOI：
10.1016/j.difgeo.2012.11.002
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发表时间：
2013
影响因子：
0.5
通讯作者：
Shigeo Kawai and Nobumitsu Nakauchi
中科院分区：
文献类型：
--
作者：
I. B´ar´any;J. Itoh;C. Vilcu;Shigeo Kawai and Nobumitsu Nakauchi
设（M,g）， （N,h）是无边界的紧黎曼流形，设f是M到N的光滑映射。我们考虑一个协变对称张量Tf=f h−1m‖df‖2g，其中f h表示度规h被f回拉，M是流形M的维数，当且仅当映射f是弱共形时张量Tf消失。范数‖Tf‖是一个度量f在每个点的一致性的量。在Nakauchi(2011)[5]中，第二作者引入了函数Φ(f)=∫M‖Tf‖2dvg的临界点映射。我们称这种映射为c -平稳映射。任何共形映射或更一般地说，任何弱共形映射都是c平稳映射。当一个c -平稳映射是一个（弱）保角映射时，我们很感兴趣。本文证明了以下结果。如果f是从标准球体Sm（m大于或等于5）或进入标准球体Sn（n大于或等于5）的稳定c -平稳映射，则f是弱保形映射。
Let (M,g), (N,h) be compact Riemannian manifolds without boundary, and let f be a smooth map from M into N. We consider a covariant symmetric tensor Tf=f⁎h−1m‖df‖2g, where f⁎h denotes the pullback of the metric h by f, and m is the dimension of the manifold M. The tensor Tfvanishes if and only if the map f is weakly conformal. The norm ‖Tf‖ is a quantity which is a measure of conformality of f at each point. In Nakauchi (2011) [5] the second author introduced maps which are critical points of the functional Φ(f)=∫M‖Tf‖2dvg. We call such maps C-stationary maps. Any conformal map or more generally any weakly conformal map is a C-stationary map. It is of interest to find when a C-stationary map is a (weakly) conformal map. In this paper we prove the following result. If f is a stable C-stationary map from the standard sphere Sm(m⩾5) or into the standard sphere Sn(n⩾5), then f is a weakly conformal map.