On Vanishing Theorems for Vector Bundle Valued p-Forms and their Applications
On Vanishing Theorems for Vector Bundle Valued p-Forms and their Applications
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DOI：
10.1007/s00220-011-1227-8
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发表时间：
2010-03
影响因子：
2.4
通讯作者：
Yuxin Dong;S. Wei
中科院分区：
文献类型：
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作者：
Yuxin Dong;S. Wei
令f:[0，∞)→[0,∞)是f(0) = 0的严格递增的c2函数。我们统一了off -调和映射、极小超曲面、极大类空间超曲面和Yang-Mills场的概念，并在流形上引入了f -Yang-Mills场、f -次、f -低次和广义Yang-Mills- born - infeld场（带正号或负号）。当和f -Yang-Mills场变成普通的Yang-Mills场时，p-Yang-Mills场分别在流形上变成带正号的广义Yang-Mills- born - infeld场和带负号的广义Yang-Mills- born - infeld场。我们还引入了ef，g−能量泛函。F-Yang-Mills泛函)，并推导出ef的第一个变分公式，g−能量泛函(参见。F-Yang-Mills功能)与应用程序。在更一般的框架下，我们使用统一的方法来研究当域或基流形的度量改变时，计算各种泛函的变化率所产生的应力-能量张量。通过黎曼几何中的协面积公式和比较定理，这些应力-能量张量自然地与守恒定律和单调性公式联系在一起。然而，对这些单调性公式中的一些的“微观”方法导致了著名的爆破技术和几何测量理论中的正则性理论，这些单调性不等式的“宏观”版本使我们能够推导出具有向量束值的p -形式的一些Liouville型结果和消失定理，并研究1-形式的常数Dirichlet边值问题。特别地，我们得到了F−调和映射（包括调和映射、p调和映射、指数调和映射、极小图和极大类空间超曲面等）、流形上的F−Yang-Mills场、扩展Born-Infeld场和广义Yang-Mills-Born-Infeld场（带正号和负号）等的Liouville定理。另一个结果是，我们得到了满足反f守恒律的向量束值1型在星形域上的常数Dirichlet边值问题的唯一常数解，推广和改进了Karcher和Wood在调和映射上的工作。我们还用另一种方法得到了具有全局加倍性质的流形上和流形上的p -型常平均曲率型方程的广义Chern型结果。casep= 0，这是由于chen。
LetF: [0, ∞) → [0, ∞) be a strictly increasingC2function withF(0) = 0. We unify the concepts ofF-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduceF-Yang-Mills fields,F-degree,F-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. WhenandtheF-Yang-Mills field becomes an ordinary Yang-Mills field,p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce theEF,g−energy functional (resp.F-Yang-Mills functional) and derive the first variational formula of theEF,g−energy functional (resp.F-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors are naturally linked toF-conservation laws and yield monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry. Whereas a “microscopic” approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory, a “macroscopic” version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems forp−forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems forF−harmonic maps (which include harmonic maps,p-harmonic maps, exponentially harmonic maps, minimal graphs and maximal space-like hypersurfaces, etc.),F−Yang-Mills fields, extended Born-Infeld fields, and generalized Yang-Mills-Born-Infeld fields (with the plus sign and with the minus sign) on manifolds, etc. As another consequence, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains for vector bundle-valued 1-forms satisfying anF-conservation law, generalizing and refining the work of Karcher and Wood on harmonic maps. We also obtain generalized Chern type results for constant mean curvature type equations forp−forms onand on manifoldsMwith the global doubling property by a different approach. The casep= 0 andis due to Chern.