## Files in this item

FilesDescriptionFormat

application/pdf

Chih-Chung_Liu.pdf (570kB)
(no description provided)PDF

## Description

 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 Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation 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 URI: http://hdl.handle.net/2142/44328 Rights Information: Copyright 2013 Chih-Chung Liu Date Available in IDEALS: 2013-05-24 Date Deposited: 2013-05
﻿