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Description

Yang-Mills connections on a vector bundle over a Riemannian 4-manifold in general need not be instantons. However, for certain manifolds there are gap theorems stating the nonexistence of non-instanton Yang-Mills connections with small (anti)-self-dual curvature. In joint work with Alex Waldron (UW-Madison), we prove a parabolic gap theorem: on the 4-sphere, the space of all connections with (anti)-self-dual energy less than a certain amount deformation retracts under Yang-Mills flow onto the space of instantons. As an application, we recover Taubes' theorem on path-connectedness of instanton moduli spaces on the 4-sphere.