Limits of Yang–Mills $${\varvec{\alpha }}$$ α -connections

Abstract

In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps (Lamm et al. in Limits of $$\alpha $$ -harmonic maps, 2015), we identify Yang–Mills connections obtained by approximations with respect to the Yang–Mills $$\alpha $$ -energy. More specifically, we show that for the $${{\,\mathrm{SU}\,}}(2)$$ Hopf fibration over $${\mathbb {S}}^4$$ , for sufficiently small $$\alpha $$ values the $${{\,\mathrm{SO}\,}}(4)$$ invariant ADHM instanton is the unique $$\alpha $$ -critical point which has Yang–Mills $$\alpha $$ -energy lower than a specific threshold.