We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Råde [21] and the bubbling criterion in dimension 4 of Struwe [23] in the case where the initial flow data is smooth.

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Following [3], we define a notion of entropy for connections over R n which has shrinking Yang-Mills solitons as critical points. As in [3], this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying "generic singularities" of Yang-Mills flow, and we discuss the differences in this strategy in dimension n = 4 versus n ≥ 5.

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Abstract. Inspired by work of Colding-Minicozzi [3] on mean curvature flow, Zhang [16] introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable F-stability. Then, focusing on the case of spherical targets, we prove a general instability result for high-entropy solitons. Finally, we exploit results of Lin-Wang [11] to observe long time existence and convergence results for maps into certain convex domains and how they relate to generic singularities of harmonic map flow.