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Uniqueness of Ricci Flow Solution on Non-compact Manifolds and Integral Scalar Curvature Bound

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dc.contributor.advisor Chen, Xiuxiong en_US
dc.contributor.author Wang, Xiaojie en_US
dc.contributor.other Department of Mathematics. en_US
dc.date.accessioned 2017-09-20T16:50:11Z
dc.date.available 2017-09-20T16:50:11Z
dc.date.issued 2014-12-01 en_US
dc.identifier.uri http://hdl.handle.net/11401/76411 en_US
dc.description 72 pg. en_US
dc.description.abstract In this dissertation, we prove two results. The first is about the uniqueness of Ricci flow solution. B.-L. Chen and X.-P. Zhu first proved the uniqueness of Ricci flow solution to the initial value problem by assuming bilaterally bounded curvature over the space-time. Here we show that, when the initial data has bounded curvature and is non-collapsing, the complex sectional curvature bounded from below over the space-time guarantees the short-time uniqueness of solution. The second is about the integral scalar curvature bound. A. Petrunin proved that for any complete boundary free Riemannian manifold, if the sectional curvature is bounded from below by negative one, then the integral of the scalar curvature over any unit ball is bounded from above by a constant depending only on the dimension. We ask whether this is true when replacing the sectional curvature with Ricci curvature in the condition. We show that, essentially, there is no counter-example with warped product metric. The application to the uniqueness of Ricci flow is also discussed. en_US
dc.description.sponsorship This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. en_US
dc.format Monograph en_US
dc.format.medium Electronic Resource en_US
dc.language.iso en_US en_US
dc.publisher The Graduate School, Stony Brook University: Stony Brook, NY. en_US
dc.subject.lcsh Mathematics en_US
dc.subject.other Differential Geometry, Integral Bound, Partial Differential Equation, Ricci Flow, Scalar Curvature, Uniqueness en_US
dc.title Uniqueness of Ricci Flow Solution on Non-compact Manifolds and Integral Scalar Curvature Bound en_US
dc.type Dissertation en_US
dc.mimetype Application/PDF en_US
dc.contributor.committeemember Anderson, Michael en_US
dc.contributor.committeemember Khuri, Marcus en_US
dc.contributor.committeemember Gu, Xianfeng. en_US

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