Browsing by Subject "Fiber bundles (Mathematics)"

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Knots on once-punctured torus fibers
(2004) Baker, Kenneth Lee; Luecke, John Edwin
We study knots that lie as essential simple closed curves on the fiber of a genus one fibered knot in S 3 . We determine certain surgery descriptions of these knots that enable estimates on volumes of these knots. We also develop an algorithm to list all closed essential surfaces in the complement of a given knot in this family. Relationships between the volumes of such knots and the surfaces in their exteriors is then examined.
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.