The analytic and asymptotic behaviors of vortices

Liu, Chih-Chung

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

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

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

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

Copyright and License Information

Copyright 2013 Chih-Chung Liu

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