The threshold conjecture for the energy critical hyperbolic Yang--Mills equation

TL;DR

This paper proves the Threshold Conjecture for the energy critical 4+1 dimensional hyperbolic Yang--Mills equation, establishing conditions for global existence, scattering, and soliton bubbling, completing a four-part research sequence.

Contribution

It finalizes the proof of the Threshold and Dichotomy Theorems for the energy critical hyperbolic Yang--Mills equation, including ruling out self-similar solutions.

Findings

Solutions with energy below twice the ground state are global and scatter.

Either solutions bubble off a soliton or do not exist as nontrivial self-similar solutions.

The proof completes the classification of solution behaviors at critical energy levels.

Abstract

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 which 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 existence of a nontrivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.