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The threshold conjecture for the energy critical hyperbolic YangMills equation
Abstract
This article represents the fourth and final part of a fourpaper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical $4+1$ dimensional hyperbolic YangMills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time. Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blowup analysis which shows that the failure of global wellposedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence existence of a nontrivial selfsimilar solution. The proof is completed by showing that the latter solutions do not exist.
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