Type
Text
Type
Dissertation
Advisor
LeBrun, Claude R. | Lawson, Blaine | Anderson, Michael | Rocek, Martin.
Date
2014-12-01
Keywords
Differential Geometry, Gauge Theory, Kahler Geometry, Seiberg-Witten Equations | Mathematics
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76392
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
Soon after the introduction of the Seiberg-Witten equations, and their magnificent application to the differential topology of 4-manifolds, LeBrun [LeB95] used these equations to study differential geometry and prove a rigidity theorem for compact complex hyperbolic manifolds. Biquard [Biq97] extended these results to non-compact, finite volume complex hyperbolic manifolds, and Rollin [Rol04] extended these techniques to CH2. Finally, Di Cerbo[DC12b, DC11] applied Biquard's techniques to the product of two negatively curved Riemann surfaces. The main tool that allows one to use the Seiberg-Witten equations to study differential geometry is an integral scalar curvature estimate The principle difficulty in extending these methods to the non-compact case, which was overcome by Biquard, Rollin and Di Cerbo is the proof of the existence of a solution to the equations. Finally, in LeBrun used conformal rescaling of the Seiberg-Witten equations to prove an integral estimate that involves both the scalar and Weyl curvature. In this thesis we extend these techniques to quasiprojective 4-manifolds which admit negatively curved, finite volume Kahler-Einstein metrics. Following Biquard's method we produce an irreducible solution to the Seiberg-Witten equations on the non-compact manifold as a limit of solutions on the compactification, and then use the Weitzenbock formula to obtain a scalar curvature estimate that is necessary for geometric applications. | 81 pages
Recommended Citation
Elson, Ilya, "Applications of the Seiberg-Witten equations to the Differential Geometry of non-compact Kahler manifolds" (2014). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2315.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2315