A compactness theorem for Fueter sections

Abstract

We prove that a sequence of Fueter sections of a bundle of compact hyperkähler manifolds $\mathfrak{X}$ over a 3-manifold $M$ with bounded energy converges (after passing to a subsequence) outside a 1-dimensional closed rectifiable subset $S\subset M$. The non-compactness along $S$ has two sources: (1) Bubbling-off of holomorphic spheres in the fibres of $\mathfrak{X}$ transverse to a subset $\Gamma \subset S$, whose tangent directions satisfy strong rigidity properties. (2) The formation of non-removable singularities in a set of ${\mathcal{H}}^1$-measure zero. Our analysis is based on the ideas and techniques that Lin developed for harmonic maps [19]. These methods also apply to Fueter sections on 4-dimensional manifolds; we discuss the corresponding compactness theorem in an appendix. We hope that the work in this paper will provide a first step towards extending the hyperkähler Floer theory developed by Hohloch, Noetzel, and Salamon [15] and Salamon [22] to general target spaces. Moreover, we expect that this work will find applications in gauge theory in higher dimensions.