# A compactness theorem for Fueter sections

### Thomas Walpuski

Massachusetts Institute of Technology, Cambridge, USA

## Abstract

We prove that a sequence of Fueter sections of a bundle of compact hyperkähler manifolds $X$ over a 3-manifold $M$ with bounded energy converges (after passing to a subsequence) outside a 1-dimensional closed rectifiable subset $S⊂M$. The non-compactness along $S$ has two sources: (1) Bubbling-off of holomorphic spheres in the fibres of $X$ transverse to a subset $Γ⊂S$, whose tangent directions satisfy strong rigidity properties. (2) The formation of non-removable singularities in a set of $H_{1}$-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.

## Cite this article

Thomas Walpuski, A compactness theorem for Fueter sections. Comment. Math. Helv. 92 (2017), no. 4, pp. 751–776

DOI 10.4171/CMH/423