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

Abstract

Given a connection $A$ on a $SU(2)$-bundle $P$ over ${\mathbb{R}}^4$ 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 $\hat{A}$ on a bundle $\hat{P}$ of different Chern class ($|c_2(A)-c_2(\hat{A})|=8{\pi }^2$), in such a way that $\hat{A}$ is gauge equivalent to $A$ in ${\mathbb{R}}^4\setminus B_{\rho }(0)$, gauge equivalent to an instanton in a smaller ball $B_{\tau \rho }(0)$, and $$YM(\hat{A})\le YM(A)+8{\pi }^2-{\epsilon }_0{\rho }^4|F_A(0){|}^2,$$ where $\tau \in (0.3,0.4)$ and ${\epsilon }_0>0$ are universal constant 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.