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dc.contributor.advisor Bray, Hubert L en_US
dc.contributor.author Cox, Graham en_US
dc.date.accessioned 2012-05-29T16:44:52Z
dc.date.available 2012-05-29T16:44:52Z
dc.date.issued 2011 en_US
dc.identifier.uri http://hdl.handle.net/10161/5699
dc.description Dissertation en_US
dc.description.abstract <p>In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere, and subsets thereof. In particular, we are interested in Min-Oo's conjecture that there exist no metrics on the upper hemisphere having scalar curvature greater or equal to that of the standard spherical metric, while satisfying certain natural geometric boundary conditions.</p><p>While the conjecture as originally stated has recently been disproved, there are still many interesting modications to consider. For instance, it has been shown that Min-Oo's rigidity conjecture holds on sufficiently small geodesic balls contained in the upper hemisphere, for metrics sufficiently close to the spherical metric. We show that this local rigidity phenomena can be extended to a larger class of domains in the hemisphere, in particular finding that it holds on larger geodesic balls, and on certain domains other than geodesic balls (which necessarily have more complicated boundary geometry). We discuss a possible method for finding the largest possible domain on which the local rigidity theorem is true, and give a Morse-theoretic interpretation of the problem.</p><p>Another interesting open question is whether or not such a rigidity statement holds for metrics that are not close to the spherical metric. We find that a scalar curvature rigidity theorem can be proved for metrics on sufficiently small geodesic balls in the hemisphere, provided certain additional geometric constraints are satisfied.</p> en_US
dc.subject Mathematics en_US
dc.subject differential geometry en_US
dc.subject geometric analysis en_US
dc.title Scalar curvature rigidity theorems for the upper hemisphere en_US
dc.type Dissertation en_US
dc.department Mathematics en_US

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