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Gluing Construction of Extensions to Finite Energy Initial Data for the Yang-Mills Equation in (3+1) Dimensions
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Effective Uniformization Theorem: A Senior Thesis
No full textIn this thesis, I present a complete proof of the Effective Uniformization Theorem, proved by Sergiu Klainerman and Jérémie Szeftel in 2019 in their work titled Effective Results on Uniformization and Intrinsic GCM Spheres.
In this exposition, I focus on proving numerous intermediate steps that might not be self-evident to the readers, shedding light on various technical results, correcting detected inaccuracies, and increasing the precision of the statements of the original paper
Finite energy global well-posedness of the (3+1)-dimensional Yang-Mills equations using a novel Yang-Mill heat flow gauge
No full textIn this thesis, we propose a novel choice of gauge for the Yang-Mills equations on the Minkowski space . A crucial ingredient is the associated Yang-Mills heat flow. Unlike the previous approaches, the new gauge is applicable for large data, while the special analytic structure of the Yang-Mills equations is still manifest.
As the first application of the new approach, we shall give new proofs of local well-posedness and finite energy global well-posedness of the Yang-Mills equations on . These are classical results first proved by S. Klainerman and M. Machedon using the method of local Coulomb gauges, which had been difficult to extend to other settings. As our approach does not possess its drawbacks (in particular the use of Uhlenbeck's lemma is avoided), it is expected to be more robust and easily applicable to other problem
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