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Title: On area comparison and rigidity involving the scalar curvature
Author: Moraru, Vlad
ISNI:       0000 0004 2749 3212
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2013
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics