The analytic and asymptotic behaviors of vortices

"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."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2013

Author dc:creator

• Liu, Chih-Chung

Contributors dc:contributor

• Bradlow, Steven B.
• Kerman, Ely
• Kirr, Eduard-Wilhelm
• La Nave, Gabriele

• Vortex Equations
• Mathematical Physics
• L^2 Geometry
• Differential Geometry.

Statement dc:rights

• Copyright 2013 Chih-Chung Liu

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/44328