Geometry of minimal energy Yang–Mills connections

where FA denotes the curvature of A. In four dimensions, FA decomposes into its self-dual and anti-self-dual components, FA = F + A + F − A , where F A denotes the projection onto the ±1 eigenspace of the Hodge star operator. A connection is called self-dual (respectively anti-self-dual) if FA = F + A (respectively FA = F − A ). A connection is called an instanton if it is either self-dual or anti-self-dual. An instanton is always a minimizer of the Yang-Mills energy on a compact oriented 4 manifold. This leads to the converse question: in four dimensions are local minima for the Yang-Mills energy necessarily instantons? The answer to this naive question has long been known to be no. (See [BLS] and [BL]). Partial positive results for low rank G, however, were obtained by Bourguignon, Lawson, and Simons in [BLS] and [BL], where they use a variational argument to show that if G = SU(2) or SU(3), and M is a compact oriented 4 dimensional homogeneous space, then the curvature, FA, is self-dual, anti-self-dual, or abelian. In this note, we settle this converse question in four dimensions for nonnegatively curved homogeneous manifolds and offer related weaker results for special geometries in higher dimensions. Our main result is the following theorem.