Type
Text
Type
Dissertation
Advisor
Lawson, Blaine | LeBrun, Claude R | Anderson, Michael | Rocek, Martin.
Date
2012-05-01
Keywords
Bach Tensor, Conformal geometry, Einstein metrics, Kahler metrics, Weyl Curvature | 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/11401/71298
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
In this thesis, we study several problems related to conformal geometry of Kahler and Einstein metrics on compact 4-manifolds, by using the conformally invariant Weyl functional. We first study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations. Inspired by the work of C. LeBrun on Einstein-Maxwell equations on compact Kahler surfaces, we give a variational characterization of solutions to Bach-Merkulov equations as critical points of the Weyl functional. We also show that extremal Kahler metrics are solutions to these equations, although, contrary to the Einstein-Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces. Next we prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is CP_2, with its Fubini-Study metric. Finally we present an alternative proof of existence of conformally compact Einstein metrics on some complex ruled surfaces fibered over Riemann surfaces of genus at least 2. This result was first proved by C. Tonnesen-Friedman. We prove the existence by finding the critical points of the Weyl functional on space of all extremal Kahler metrics on these ruled surfaces. | 43 pages
Recommended Citation
Koca, Caner, "On Conformal Geometry of Kahler Surfaces" (2012). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 504.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/504