- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Non-existence of metrics with positive scalar curvature...
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Non-existence of metrics with positive scalar curvature on the n-torus Friedman, Benjamin
Abstract
In this thesis, we explore a famous theorem of Schoen and Yau stating that there exists no metric of positive scalar curvature on the n-torus Tⁿ, for n ≤ 7. The proof is by induction: One first assumes a metric of positive scalar curvature on Tⁿ. Then, by applying techniques from geometric measure theory, the direct method of the calculus of variations yields an area minimizing hypersurface in each non-trivial homology class χ ∈ Hn-1(Tⁿ; ℤ). Using a stability argument, it is then shown that the induced metric on this hypersurface is conformal to a metric of positive scalar curvature, contradicting the inductive assumption.
Item Metadata
| Title |
Non-existence of metrics with positive scalar curvature on the n-torus
|
| Creator | |
| Supervisor | |
| Publisher |
University of British Columbia
|
| Date Issued |
2022
|
| Description |
In this thesis, we explore a famous theorem of Schoen and Yau stating that there exists no metric of positive scalar curvature on the n-torus Tⁿ, for n ≤ 7. The proof is by induction: One first assumes a metric of positive scalar curvature on Tⁿ. Then, by applying techniques from geometric measure theory, the direct method of the calculus of variations yields an area minimizing hypersurface in each non-trivial homology class χ ∈ Hn-1(Tⁿ; ℤ). Using a stability argument, it is then shown that the induced metric on this hypersurface is conformal to a metric of positive scalar curvature, contradicting the inductive assumption.
|
| Genre | |
| Type | |
| Language |
eng
|
| Date Available |
2022-04-21
|
| Provider |
Vancouver : University of British Columbia Library
|
| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
| DOI |
10.14288/1.0412956
|
| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
|
| Graduation Date |
2022-05
|
| Campus | |
| Scholarly Level |
Graduate
|
| Rights URI | |
| Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International