Higher order Yang–Mills flow

Abstract

We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Råde [21] and the bubbling criterion in dimension 4 of Struwe [23] in the case where the initial flow data is smooth.

Higher order Seiberg–Witten functionals and their associated gradient flows

“…The above theorem shows that in the critical dimension, long time existence is obstructed by the possibility of the curvature form concentrating in smaller and smaller balls. This is analogous to what Struwe observed for the Yang-Mills flow in dimension four (see theorem 2.3 in [18]), and what Kelleher observed for the higher order Yang-Mills flow in the critical dimension (see theorem B in [8]).…”

“…Furthermore, when n = 2(k + 2) we cannot rule out finite time singularities, but we show that if present, they are due to an L k+2 curvature concentration phenomenon, see proposition 8.3 and theorem 8.4. This is analogous to what Kelleher observes for the higher order Yang-Mills flow (theorem B [8]). However, this is in contrast with the work of Hong and Schabrun (theorem 1 in [5]) and Schabrun (theorem 1 in [15]), on the Seiberg-Witten flow, who are able to show that an L 2 curvature concentration phenomenon can obstruct long time existence, but are able to rule out such concentration by a careful rescaling argument together with an L 2 energy estimate.…”

“…The above theorem can be seen as an analogue of the first part of theorem 7.8 in [10], and theorem A in [8], for the case of these higher order Seiberg-Witten functionals.…”

“…We will now show that in the case that the curvature form is blowing up, as one approaches the maximal time, a blow up limit can be extracted. The proof of the theorem will closely follow the proof of proposition 3.24 in [8], and the proof of lemma 4.6 in [5].…”

“…He is able to show (theorem 7.8 [10]), that provided the derivatives in his functionals are large enough, singularities in finite time do not occur. Inspired by this, Kelleher in [8] considers a higher order variant of the Yang-Mills flow. She proves long time existence in sub-critical dimensions (theorem A in [8]), and is able to prove a curvature concentration phenomenon in the critical dimension (theorem B [8]), analogous to the result obtained by Struwe (theorem 2.3 [18]) for the Yang-Mills flow in dimension four.…”