Biharmonic Maps Between Riemannian Manifolds
Biharmonic Maps Between Riemannian Manifolds
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DOI：
10.1142/9789811212383_0010
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
R. Caddeo;E. Loubeau;S. Montaldo;C. Oniciuc;M. Piu
中科院分区：
文献类型：
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作者：
R. Caddeo;E. Loubeau;S. Montaldo;C. Oniciuc;M. Piu
点的双能泛函E2（'）= 1 R M j？（'）j 2 vg; where？（'）是'的张力场。双调和映射是调和映射的一个自然扩展（？（'）= 0）。虽然E2自60年代初以来一直在数学舞台上，当它的一些分析方面已经被讨论，其临界点的规律性现在是一个发展良好的领域，但对双调和映射几何的系统研究直到最近才开始。本文主要讨论了双调和映射的几何性质，并介绍了这方面的一些最新成果：（a）给出了某些Thurston几何的双调和曲线曲面的显式分类[2，3，4]。(b)本文描述了翘曲积间映射的双调和性，并利用这一条件研究了三类轴对称双调和映射[1]。(c)利用Hilbert准则，我们考虑了与双能量相关的应力-能量张量，证明了它是由度量的变分问题导出的，并展示了四维的特殊性，利用应力-能量张量构造了双调和映射的新例子[5]。
points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the mathematical scene since the early ’60, when some of its analytical aspects have been discussed, and regularity of its critical points is nowadays a well-developed fleld, a systematic study of the geometry of biharmonic maps has started only recently. In this lecture we focus on the geometric properties of biharmonic maps and describe some recent achievements on the subject: (a) We give the explicit classiflcations of biharmonic curves and surfaces of some Thurston’s geometries [2, 3, 4]. (b) We describe the biharmonicity of maps between warped products and using this setting we study three classes of axially symmetric biharmonic maps [1]. (c) Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy, show it derives from a variational problem on metrics, exhibit the peculiarity of dimension four, and use the stress-energy tensor to construct new examples of biharmonic maps [5].