On bubbling of Fueter maps

Abstract

We study bubbling for sequences of solutions of the Hamiltonian-perturbed Fueter equation. By Walpuski's compactness theorem, a bounded-energy sequence converges weakly to a Fueter map, with energy loss recorded by a defect measure supported on a codimension-two rectifiable set. Assuming that the limiting map has no non-removable singularities, the bubbling locus contains a nontrivial Lipschitz arc and the bubbles attach to the limiting map along this arc, we show that the limiting Hamiltonian lies in an exceptional subset of infinite codimension. The proof combines quantitative control of rational sweepout loci in hyperk\"ahler manifolds with a transversality argument. We expect both the Lipschitz-arc and attachment assumptions to be automatic. This strongly suggests that, in the absence of non-removable singularities, bubbling of Hamiltonian-perturbed Fueter maps is highly nongeneric.