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Commentarii Mathematici Helvetici

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Volume 92, Issue 4, 2017, pp. 751–776
DOI: 10.4171/CMH/423

Published online: 2017-10-24

A compactness theorem for Fueter sections

Thomas Walpuski

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.

Keywords: Fueter sections, compactness, bubbling, hyperkähler manifolds

Walpuski Thomas: A compactness theorem for Fueter sections. Comment. Math. Helv. 92 (2017), 751-776. doi: 10.4171/CMH/423