Isolation phemonena for quaternionic Yang-Mills connections (Q5906928)

Let M be a compact Riemannian 4-manifold. The Hodge star operator is an involution which gives rise to the notions of self-dual and anti-selfdual 2-forms. A connection in a vector bundle over M is called (anti-)self-dual if its curvature is (anti-)self-dual. It is well known that (anti-)self-dual connections minimise the Yang-Mills (YM) functional. Bourguignon and Lawson discovered the following isolation phenomenon. Let \(\nabla\) be a connection in a vector bundle over the 4-sphere and denote by \(F^+\) the self-dual part of its curvature. If the inequality \(|F^+|^2<3\) is satisfied pointwise then \(F^+=0\). Of course, the same is true for \(F^-\). The author generalizes the results of Bourguignon and Lawson from the 4-sphere to arbitrary Wolf-spaces. Recall that a Wolf-space is a compact quaternionic Kähler symmetric space of positive scalar curvature. These spaces are precisely the known compact quaternionic Kähler manifolds of positive scalar curvature. These are Riemannian manifolds of dimension \(4n\) with holonomy group in \(\text{Sp}(1)\text{Sp}(n)\), and have arbitrary dimension divisible by 4. Nevertheless, the notion of (anti-)self-dual 2-form can be generalised to such manifolds, see the papers of Nitta, Mamone Capria and Salamon cited in the paper under review. In fact, one can define three (not only two) types of generalized self-dual 2-forms corresponding to the three irreducible \(\text{Sp}(1)\text{Sp}(n)\)-submodules in the vector space of 2-forms. Forms belonging to one of the submodules which correspond to the Lie algebras of \(\text{Sp}(1)\) and \(\text{Sp}(n)\) are usually called (anti-)self-dual 2-forms. With this definition, (anti-)self-dual connections again minimise the YM functional on compact quaternionic Kähler manifolds.