Local monotonicity for the Yang–Mills–Higgs flow

Abstract

A local monotonicity formula for the Yang-Mills-Higgs flow on G-bundles over R n (n > 4) is proved. It is shown that the monotone quantity coïncides on certain selfsimilar solutions with that appearing in existing non-local monotonicity formulae for the Yang-Mills and Yang-Mills-Higgs flows.

Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

“…where φ (x,t) (y, s) = r(x,y) 2 4(s−t) − n−2k 2 log (4π(t − s)). Note that the right-hand side implicitly depends on the choice of metric g. The following was established by Ecker [14] in the case of the harmonic map heat flow with M Euclidean and more generally for the Yang-Mills and harmonic map heat flows on Riemannian manifolds by the author [1,3]. is monotone nondecreasing for r < r 0 whenever the integrand is defined and summable over E n−2k r 0…”

“…It was shown in [3] that the quantity (2.19) is finite for r ≤ r 0 whenever 1 2 |ψ| 2 is summable over a suitable parabolic cylinder containing E n−2k r 0 (x, t). In fact, we have the following estimate.…”

“…While it is possible to introduce a cut-off function in (1.1) to prove an analogue of Theorem A for radii smaller than a constant depending on geometry and the local energy of u as has been done in [23] for the Yang-Mills-Higgs flow, our approach works out more cleanly and closely parallels the analogous regularity results in the static case, which are stated in terms of suitable rescaled energies on balls (see [32,Proposition 2.4] and [32,Theorem 3.1] for the case of harmonic maps as well as [30,Theorem 1] and [26,Lemma 3.1] for the case of Yang-Mills connections). Furthermore, our proof rests on a fairly simple blow-up argument as well as suitable local monotonicity formulae established by Ecker [14] for the harmonic map heat flow on Euclidean domains and the author [1,3] more generally for both the harmonic map and Yang-Mills heat flows on Riemannian manifolds. The blow-up argument we use was employed by White [42] in establishing a similar local regularity result for the mean curvature flow (see also [13,Chap.…”

A new higher order Yang--Mills--Higgs flow in Riemannian $4$-manifold

“…Hong-Tian studied the global existence of Yang-Mills-Higgs flow in 3-dimensional Hyperbolic space, their results yields non-self dual Yang-Mills connections on S 4 . In the new century, the study of Yang-Mills-Higgs flow has aroused a lot of attention (see [1,6,9,11,12,15,16] and references therein).…”

Gradient Flows of Higher Order Yang-Mills-Higgs Functionals

“…The study of Yang-Mills-Higgs flow has aroused a lot attention in the new century (see [1,13,14,17,19,20,29,32,45,49,56,57,58] and so on). In spite of the work of Waldron [53], it is natural to ask: do the finite-time singularities occur in 4-dimensional Yang-Mills-Higgs flow?…”