2015/07/31 by Thomas Walpuski · 1 citation
Mathematics · #math.DG #msc:53C26 #msc:53C43 #msc:58E20
paper · pdf · doi:10.4171/cmh/423
published as Commentarii Mathematici Helvetici, Volume 92, Issue 4, pp. 751-776 (2017) · v2: published version
arxiv created 2018/10/01 · arxiv updated 2018/10/02
We prove that a sequence of Fueter sections of a bundle of compact hyperkahler 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 ⊂ 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 Γ⊂ S, whose tangent directions satisfy strong rigidity properties. (2) The formation of non-removable singularities in a set of \mathcal H1-measure zero. Our analysis is based on the ideas and techniques that Lin developed for harmonic maps. 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 hyperkahler Floer theory developed by Hohloch-Noetzl-Salamon to general target spaces. Moreover, we expect that this work will find applications in gauge theory in higher dimensions.