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.

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We establish a local monotonicity identity for vector bundle-valued differential k-forms on superlevel sets of appropriate heat kernel-like functions. As a consequence, we obtain new local monotonicity formulæ for the harmonic map and Yang–Mills heat flows on evolving manifolds. We also show how these methods yield local monotonicity formulæ for the Yang–Mills–Higgs flow.

We investigate monotonicity properties of p-harmonic vector bundle-valued k-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formulae for p-harmonic maps and Yang-Mills connections, proving a monotonicity formula for p-Yang-Mills connections in the process. Moreover, it is shown how this technique may be adapted to yield an analogous monotonicity formula for Yang-Mills-Higgs pairs. Finally, we obtain Liouville-type theorems for such forms and Yang-Mills-Higgs pairs as an application.

We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).