Discreteness for energies of Yang-Mills connections over four-dimensional manifolds

We generalize our previous results (Theorem 1 and Corollary 2 in arXiv:1412.4114) and Theorem 1 in arXiv:1502.00668) on the existence of an $L^2$-energy gap for Yang-Mills connections over closed four-dimensional manifolds and energies near the ground state (occupied by flat, anti-self-dual, or self-dual connections) to the case of Yang-Mills connections with arbitrary energies. We prove that for any principal bundle with compact Lie structure group over a closed, four-dimensional, Riemannian manifold, the $L^2$ energies of Yang-Mills connections on a principal bundle form a discrete sequence without accumulation points. Our proof employs a version of our {\L}ojasiewicz-Simon gradient inequality for the Yang-Mills $L^2$-energy functional from our monograph arXiv:1409.1525 and extensions of our previous results on the bubble-tree compactification for the moduli space of anti-self-dual connections arXiv:1504.05741 to the moduli space of Yang-Mills connections with a uniform $L^2$ bound on their energies.