18 research outputs found

    On Vanishing Theorems For Vector Bundle Valued p-Forms And Their Applications

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    Let F:[0,)[0,)F: [0, \infty) \to [0, \infty) be a strictly increasing C2C^2 function with F(0)=0F(0)=0. We unify the concepts of FF-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce FF-Yang-Mills fields, FF-degree, FF-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When F(t)=t,1p(2t)p2,1+2t1,F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1, and 112t,1-\sqrt{1-2t}, the FF-Yang-Mills field becomes an ordinary Yang-Mills field, pp-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 the EF,gE_{F,g}-energy functional (resp. FF-Yang-Mills functional) and derive the first variational formula of the EF,gE_{F,g}-energy functional (resp. FF-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, linked to FF-conservation laws yield monotonicity formulae. A "macroscopic" version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for pp-forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for FF-harmonic maps (e.g. pp-harmonic maps), and FF-Yang-Mills fields (e.g. generalized Yang-Mills-Born-Infeld fields on manifolds). We also obtain generalized Chern type results for constant mean curvature type equations for pp-forms on Rm\Bbb{R}^m and on manifolds MM with the global doubling property by a different approach. The case p=0p=0 and M=RmM=\mathbb{R}^m is due to Chern.Comment: 1. This is a revised version with several new sections and an appendix that will appear in Communications in Mathematical Physics. 2. A "microscopic" approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory. 3. Our unique solution of the Dirichlet problems generalizes the work of Karcher and Wood on harmonic map
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