Harmonic Maps, Rigidity, and Hodge Theory
Harmonic Maps, Rigidity, and Hodge Theory
复制标题
调和图、刚度和霍奇理论
DOI：
10.1007/978-3-0348-9078-6_39
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
K. Corlette
中科院分区：
文献类型：
--
作者：
K. Corlette
调和映射是调和函数的非线性类似物，如果考虑它们的微分，就是调和1形式。因此，我们可以期待霍奇理论关于谐波形式的类似结果。调和映射是两个黎曼流形之间映射上的能量泛函的临界点。如果M， n黎曼流形和f:M→n黎曼流形之间的光滑映射，则能量由f的微分处定义。如果它的能量有限，那么我们可以问它是否为临界点；对应的欧拉-拉格朗日方程inD*df= 0，其中，ed是与off* tnn自然连接相关联的外导数算子，而dfi被视为1-form的onm，其值为inf*TN。后者是调和映射方程。它是拉普拉斯方程的非线性模拟。
Harmonic maps are nonlinear analogues of harmonic functions or, if one considers their differentials, harmonic 1-forms. As such, one can expect analogues of Hodge-theoretic results about harmonic 1-forms. Harmonic maps arise as critical points for the energy functional on maps between two Riemannian manifolds. IfM, Nare Riemannian manifolds andf:M→Nis a smooth map between them, then the energy is defined bywheredfis the differential off. Iffhas finite energy, then we can ask whetherfis a critical point forE; the corresponding Euler-Lagrange equation inD*df= 0, whereDis the exterior derivative operator associated to the natural connection off*TNanddfis regarded as a 1-form onMwith values inf*TN. The latter is the harmonic map equation. It is nonlinear analogue of Laplace’s equation.