GRADIENT FLOWS OF HIGHER ORDER YANG-MILLS-HIGGS FUNCTIONALS

In this paper, we define a family of functionals generalizing the Yang-Mills-Higgs functionals 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 a maximum 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 four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang-Mills-Higgs functionals is strictly greater than one, 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), for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local L-2-derivative estimates, energy estimates and blow-up analysis.