University of Illinois Urbana-Champaign

The analytic and asymptotic behaviors of vortices

Liu, Chih-Chung

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Chih-Chung_Liu.pdf

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https://hdl.handle.net/2142/44328

Description

Title
The analytic and asymptotic behaviors of vortices
Author(s)
Liu, Chih-Chung
Issue Date
2013-05-24T22:07:57Z
Director of Research (if dissertation) or Advisor (if thesis)
Bradlow, Steven B.
Doctoral Committee Chair(s)
  • Kerman, Ely
  • Kirr, Eduard-Wilhelm
Committee Member(s)
  • Bradlow, Steven B.
  • La Nave, Gabriele
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2013-05-24T22:07:57Z
Keyword(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."
Graduation Semester
2013-05
Permalink
http://hdl.handle.net/2142/44328
Copyright and License Information
Copyright 2013 Chih-Chung Liu

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