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The threshold conjecture for the energy critical hyperbolic Yang-Mills equation. (English) Zbl 1473.35071

Summary: This article represents the fourth and final part of a four-paper 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 Yang-Mills 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 blow-up analysis that shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence of a non-trivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

MSC:

35B44 Blow-up in context of PDEs
35C06 Self-similar solutions to PDEs
35L72 Second-order quasilinear hyperbolic equations
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
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