{"@context":"http://iiif.io/api/presentation/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/manifest.json","@type":"sc:Manifest","label":"Uniqueness of Ricci Flow Solution on Non-compact Manifolds and Integral Scalar Curvature Bound","metadata":[{"label":"dc.description.sponsorship","value":"This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree."},{"label":"dc.format","value":"Monograph"},{"label":"dc.format.medium","value":"Electronic Resource"},{"label":"dc.identifier.uri","value":"http://hdl.handle.net/11401/76411"},{"label":"dc.language.iso","value":"en_US"},{"label":"dc.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.abstract","value":"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."},{"label":"dcterms.available","value":"2017-09-20T16:50:11Z"},{"label":"dcterms.contributor","value":"Gu, Xianfeng."},{"label":"dcterms.creator","value":"Wang, Xiaojie"},{"label":"dcterms.dateAccepted","value":"2017-09-20T16:50:11Z"},{"label":"dcterms.dateSubmitted","value":"2017-09-20T16:50:11Z"},{"label":"dcterms.description","value":"Department of Mathematics."},{"label":"dcterms.extent","value":"72 pg."},{"label":"dcterms.format","value":"Monograph"},{"label":"dcterms.identifier","value":"http://hdl.handle.net/11401/76411"},{"label":"dcterms.issued","value":"2014-12-01"},{"label":"dcterms.language","value":"en_US"},{"label":"dcterms.provenance","value":"Made available in DSpace on 2017-09-20T16:50:11Z (GMT). No. of bitstreams: 1\nWang_grad.sunysb_0771E_12038.pdf: 354933 bytes, checksum: b2b48f3570e45fe6d08045c2b7fa6a0b (MD5)\n  Previous issue date:    1"},{"label":"dcterms.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.subject","value":"Differential Geometry, Integral Bound, Partial Differential Equation, Ricci Flow, Scalar Curvature, Uniqueness"},{"label":"dcterms.title","value":"Uniqueness of Ricci Flow Solution on Non-compact Manifolds and Integral Scalar Curvature Bound"},{"label":"dcterms.type","value":"Dissertation"},{"label":"dc.type","value":"Dissertation"}],"description":"This manifest was generated dynamically","viewingDirection":"left-to-right","sequences":[{"@type":"sc:Sequence","canvases":[{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json","@type":"sc:Canvas","label":"Page 1","height":1650,"width":1275,"images":[{"@type":"oa:Annotation","motivation":"sc:painting","resource":{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/66%2F82%2F77%2F66827739598049646443264424955903590510/full/full/0/default.jpg","@type":"dctypes:Image","format":"image/jpeg","height":1650,"width":1275,"service":{"@context":"http://iiif.io/api/image/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/66%2F82%2F77%2F66827739598049646443264424955903590510","profile":"http://iiif.io/api/image/2/level2.json"}},"on":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json"}]}]}]}