Browsing by Subject "Yang-Mills theory"
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Item Adiabatic limits of the Hermitian Yang-Mills equations on slicewise stable bundles(2002) Mandolesi, André Luís Godinho; Uhlenbeck, Karen K.A formal limit of the Hermitian Yang-Mills Equations on a SU(2) bundle over a product of two Riemann surfaces yields the Adiabatic Equations when the metric of the first surface is stretched ad infinitum. This thesis identifies the solutions of this new set of equations with holomorphic maps from the first surface into the moduli space of flat connections of the second one. Moreover, some advance is made in the study of the sort of bubbling phenomena that may occur when taking this limit. This dissertation is a step towards a rigorous proof of the relationship suggested by Bershadky, Johansen, Sadov and Vafa between Donaldson invariants and quantum cohomology, and relates to the program of Dostoglou and Salamon to prove the Atiyah-Floer conjecture.Item Modified Ricci flow on a principal bundle(2008-05) Young, Andrea Nicole, 1979-; Uhlenbeck, Karen K.Let M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow. In this thesis, we derive the Ricci Yang-Mills flow and show that solutions exist for a short time and are unique. We study obstructions to the long-time existence of the flow and prove a compactness theorem for pointed solutions. We represent the Ricci Yang-Mills flow as a gradient flow and derive monotonicity formulas that can be used to study breather and soliton solutions. Finally, we use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the Ricci Yang-Mills flow in dimension 2 at Einstein Yang-Mills metrics.