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On area comparison and rigidity involving the scalar curvature
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Moraru, Vlad (2013) On area comparison and rigidity involving the scalar curvature. PhD thesis, University of Warwick.
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WRAP_THESIS_Moraru_2013.pdf - Submitted Version Download (560Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2688258~S1
Abstract
In this thesis we study the effects of lower bounds for the curvature of a Riemannian
manifold M on the geometry and topology of closed, minimal hypersurfaces. We will
prove an area comparison theorem for totally geodesic surfaces which is an optimal
analogue of the Heintze-Karcher-Maeada Theorem in the context of 3-manifolds
with lower bounds on scalar curvature (Theorem 3.8). The optimality of this result
will be addressed by explicitly constructing several counterexamples in dimensions
n ≥ 4. This area comparison theorem turns out that it provides a unified proof of
three splitting and rigidity theorems for 3-manifolds with lower bounds on the scalar
curvature that were first proved, independently, by Cai-Galloway, Bray-Brendle-
Neves and Nunes (Theorem 4.7 (a)-(c)). In the final part of this thesis we will address
some natural higher dimensional generalisations of these splitting and rigidity results
and emphasise some connections with the Yamabe problem.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Riemannian manifolds, Scalar field theory, Curvature | ||||
Official Date: | May 2013 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Micallef, Mario | ||||
Extent: | v, 67 leaves | ||||
Language: | eng |
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