On the Existence and Regularity Theory of Yang-Mills Fields

This work investigates two regularization techniques designed for identifying critical points of the Yang-Mills energy.In the first half of the dissertation, we define a family of higher order functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for critical dimensions. Consequently, we generalize the results of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Rade and the bubbling criterion in dimension 4 of Struwe in the case where the initial flow data is smooth. This encompasses the contents of the author’s paper solo paper from 2014.In the second half of the dissertation we study an alternate type of regularization. In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over S4, for sufficiently small α values the SO(4)-invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold. This is an overview of the author’s solo paper from 2016