Gradient Flows of Higher Order Yang–mills–higgs Functionals

Abstract

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.

A New Higher Order Yang–mills–higgs Flow on Riemannian -Manifolds

“…We can use De Turck's trick to establish the local existence of the Yang-Mills-Higgs k-flow. We refer to [6,9,13] for more details. As the proof is standard, we will omit the details.…”

“…From an analytic point of view, the Yang-Mills-Higgs k-flow (1.2) admits similar properties to the case in which the Higgs field takes values in Ω 0 (E). In fact, by the approach in [13], we can prove the following theorem. THEOREM 1.1.…”

“…where k ∈ N ∪ {0}, ∇ is a connection on E and u is a section of Ω 0 (adE). In [13], we considered the case when u is a section of Ω 0 (E). When k = 0, (1.1) is the Yang-Mills-Higgs functional studied in [4,5].…”

“…We consider the Yang–Mills–Higgs k -functional (or Yang–Mills–Higgs k -energy): where , is a connection on E and u is a section of . In [13], we considered the case when u is a section of . When , (1.1) is the Yang–Mills–Higgs functional studied in [4, 5].…”

“…An interesting aspect of this work is that by using a different blow-up procedure, we are able to obtain a proof of Theorem 1.2, which may be of independent interest. Another interesting aspect is that the proof of long-time existence obstruction (see Theorem 3.7) relies on properties of the Green function, which is very different from the previous techniques in [6, 9, 13].…”