论文标题
高阶Yang-Mills-higgs功能的梯度流动
Gradient Flows of Higher Order Yang-Mills-Higgs Functionals
论文作者
论文摘要
在本文中，我们定义了一个在封闭的riemannian歧管上起作用的Yang-Mills-higgs的功能家族。然后，我们通过量规固定技术证明了相应梯度流的短时间存在。高阶操作员缺乏最大原则，在Higgs领域的估计中给我们带来了很多不便。我们观察到，Higgs字段的$ L^2 $结合足以满足$ 4 $尺寸的能源估计，我们表明，如果衍生物的顺序出现，以更高的Yang-Mills-Higgs功能出现，则严格大于1的梯度流量，对梯度流的解决方案不会击中任何有限的时间。至于Yang-Mills-higgs $ k $ - 功能与希格斯自我交流，我们表明，提供了$ \ dim（m）<2（k+1）$，相关的梯度流允许使用平稳的初始数据的长期存在。证明取决于本地$ l^2 $衍生估计，能量估计和爆炸分析。
In this paper, we define a family of functionals generalizing the Yang-Mills-Higgs functional on a closed Riemannian manifold. Then we prove the short time existence of the corresponding gradient flow by a gauge fixing technique. The lack of maximal principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$-bound of the Higgs field is enough for energy estimates in $4$ dimension, and we show that, provided the order of derivatives, appearing in the higher order Yang-Mills-Higgs functionals, is strictly greater than 1, solutions to the gradient flow do not hit any finite time singularities. As for the Yang-Mills-Higgs $k$-functional with Higgs self-interaction, we show that, provided $\dim(M)<2(k+1)$, the associated gradient flow admits long time existence with smooth initial data. The proof depends on local $L^2$-derivative estimates, energy estimates and blow-up analysis.
