A compactness theorem for Fueter sections

TL;DR

This paper establishes a compactness theorem for Fueter sections over 3-manifolds, describing their convergence properties and singularity formation, with implications for hyperkähler Floer theory and higher-dimensional gauge theory.

Contribution

It proves a new compactness result for Fueter sections, detailing the nature of non-compactness and singularities, extending techniques from harmonic maps to this setting.

Findings

Convergence of Fueter sections outside a 1D rectifiable set

Identification of bubbling-off phenomena involving holomorphic spheres

Analysis of non-removable singularities with measure zero

Abstract

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 $S \subset M$. 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. 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.