Gluing instantons à la Brezis–Coron in dimension four and the dipole construction

Given a connection A on a SU(2)-bundle P over R^4 with finite Yang–Mills energy YM(A) and nonzero curvature F_A(0) at the origin, and given \rho > 0 small enough, we construct a new connection A_\rho on a bundle P of different Chern class (c2(A)−c2(A_\rho) = 8π^2), in such a way that A_\rho is gauge equivalent to A in R^4 \ B_\rho (0), gauge equivalent to an instanton in a smaller ball B_{\tau\rho} (0), and YM(A_\rho) ≤ YM(A)+8π2 −\eps_0\rho^4 |F_A(0)|^2 , where \tau \in (0.3,0.4) and \eps_0 > 0 are universal constants independent of A and \rho. Our gluing method is similar in spirit to the one of Brezis–Coron for harmonic maps. We compare it with classical results by Taubes and discuss applications and open problems.