Gradient Flows of Higher Order Yang-Mills-Higgs Functionals

Abstract

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.

22 pages

References in corpus (4)

• Surface diffusion flow near spheres
• Long-time existence for Yang-Mills flow
• Higgs bundles over non-compact Gauduchon manifolds
• Gradient flow of the norm squared of a moment map over Kahler manifolds