THE THRESHOLD THEOREM FOR THE (4+1)-DIMENSIONAL YANG-MILLS EQUATION: AN OVERVIEW OF THE PROOF

This article is devoted to the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers.