Type
Text
Type
Dissertation
Advisor
Lawson, Blaine | Anderson, Michael | LeBrun, Claude | Grigorian, Sergey.
Date
2015-08-01
Keywords
Calibrated geometry, conformal change, mass-minimizing | 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/11401/76417
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
This thesis is concerned with the question: given a submanifold (perhaps with singularities), when is it possible to change the metric in some specific way so that the submanifold becomes homologically mass-minimizing? \\ We studied this question for ``horizontal" change of metrics and conformal change of metrics for both compact and non-compact submanifolds. We also explored cases with singularities or boundaries. The main idea is to use the theory of calibrated geometry and gluing techniques. \\ As a special case, we confirm that any given oriented compact connected submanifold which is not $\mathbb{R}$-homologous to zero in a Riemannian manifold can be calibrated after a highly controlled conformal change of the given metric. The statement remains true for any non-connected submanifold as well provided the convex hull of $\mathbb{R}$-homology classes represented by its oriented connected components does not contain zero. | 97 pages
Recommended Citation
Zhang, Yongsheng, "Gluing Techniques in Calibrated Geometry" (2015). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2340.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2340