The analytic and asymptotic behaviors of vortices

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

Liu, Chih-Chung

关键词: Vortex Equations;    Mathematical Physics;    L^2 Geometry;    Differential Geometry.;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/44328/Chih-Chung_Liu.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 differentialequations 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.