A compactness theorem for Fueter sections

  • Thomas Walpuski

    Massachusetts Institute of Technology, Cambridge, USA


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