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Title:The analytic and asymptotic behaviors of vortices
Author(s):Liu, Chih-Chung
Director of Research:Bradlow, Steven B.
Doctoral Committee Chair(s):Kerman, Ely; Kirr, Eduard-Wilhelm
Doctoral Committee Member(s):Bradlow, Steven B.; La Nave, Gabriele
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Vortex Equations
Mathematical Physics
L^2 Geometry
Differential Geometry.
Abstract:We study vortex equations with a parameter $s$ on smooth vector bundles $E$ over compact K\"ahler manifolds $M$. For each $s$, we invoke techniques in \cite{Br} by turning vortex equations into the elliptic partial differential equations considered in \cite{kw} and obtain a family of solutions. Our results show that away from a singular set, such a family exhibit well controlled convergent behaviors, leading us to prove conjectures posed by Baptista in \cite{Ba} concerning dynamic behaviors of vortices. These results are published in \cite{Li}. We also analyze the analytic singularities on the singular set. The analytic singularities of the PDE's reflect topological inconsistencies as $s \to \infty$. On the second part of the thesis, we form a modification of the limiting objects, leading to a phenomenon of energy concentration known as the "bubbling". We briefly survey the established bubbling results in literature.
Issue Date:2013-05-24
Rights Information:Copyright 2013 Chih-Chung Liu
Date Available in IDEALS:2013-05-24
Date Deposited:2013-05

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