A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature
We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular poin...
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| creator | Buzano, Reto Di Matteo, Gianmichele |
| description | We develop a refined singularity analysis for the Ricci flow by investigating
curvature blow-up rates locally. We first introduce general definitions of Type
I and Type II singular points and show that these are indeed the only possible
types of singular points. In particular, near any singular point the Riemannian
curvature tensor has to blow up at least at a Type I rate, generalising a
result of Enders, Topping and the first author that relied on a global Type I
assumption. We also prove analogous results for the Ricci tensor, as well as a
localised version of Sesum's result, namely that the Ricci curvature must blow
up near every singular point of a Ricci flow, again at least at a Type I rate.
Finally, we show some applications of the theory to Ricci flows with bounded
scalar curvature. |
| doi_str_mv | 10.48550/arxiv.2006.16227 |
| format | Article |
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curvature blow-up rates locally. We first introduce general definitions of Type
I and Type II singular points and show that these are indeed the only possible
types of singular points. In particular, near any singular point the Riemannian
curvature tensor has to blow up at least at a Type I rate, generalising a
result of Enders, Topping and the first author that relied on a global Type I
assumption. We also prove analogous results for the Ricci tensor, as well as a
localised version of Sesum's result, namely that the Ricci curvature must blow
up near every singular point of a Ricci flow, again at least at a Type I rate.
Finally, we show some applications of the theory to Ricci flows with bounded
scalar curvature.</description><identifier>DOI: 10.48550/arxiv.2006.16227</identifier><language>eng</language><subject>Mathematics - Analysis of PDEs ; Mathematics - Differential Geometry</subject><creationdate>2020-06</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2006.16227$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2006.16227$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Buzano, Reto</creatorcontrib><creatorcontrib>Di Matteo, Gianmichele</creatorcontrib><title>A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature</title><description>We develop a refined singularity analysis for the Ricci flow by investigating
curvature blow-up rates locally. We first introduce general definitions of Type
I and Type II singular points and show that these are indeed the only possible
types of singular points. In particular, near any singular point the Riemannian
curvature tensor has to blow up at least at a Type I rate, generalising a
result of Enders, Topping and the first author that relied on a global Type I
assumption. We also prove analogous results for the Ricci tensor, as well as a
localised version of Sesum's result, namely that the Ricci curvature must blow
up near every singular point of a Ricci flow, again at least at a Type I rate.
Finally, we show some applications of the theory to Ricci flows with bounded
scalar curvature.</description><subject>Mathematics - Analysis of PDEs</subject><subject>Mathematics - Differential Geometry</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNpNz71OwzAYhWEvDKhwAUx8N5BgJ_5JxxBRQKqERLtHX_xDLZkkOE5L7h4oDExnOXqlh5AbRnNeCUHvMH76Y15QKnMmi0Jdko8atoPGADvfv80Bo08L1D2GZfITuCFCOlh49Vp72IThBNgb8GmCehyD15j80E-Qhn-XCU4-HeB-mHtjDey-6xihmeMR0xztFblwGCZ7_bcrst887JunbPvy-NzU2wylUpmwhmuhqnWlHOVWuM6sC1lQRGSu47rTHTKKXKlSlNQxpizvWMnQUCmVFeWK3P5mz-Z2jP4d49L-2NuzvfwCtLZVFw</recordid><startdate>20200629</startdate><enddate>20200629</enddate><creator>Buzano, Reto</creator><creator>Di Matteo, Gianmichele</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20200629</creationdate><title>A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature</title><author>Buzano, Reto ; Di Matteo, Gianmichele</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a677-5ed4c578987f04e5fbd92620aaa1fb4cbcba10a4773530f117e4b131ad0667e53</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Mathematics - Analysis of PDEs</topic><topic>Mathematics - Differential Geometry</topic><toplevel>online_resources</toplevel><creatorcontrib>Buzano, Reto</creatorcontrib><creatorcontrib>Di Matteo, Gianmichele</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Buzano, Reto</au><au>Di Matteo, Gianmichele</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature</atitle><date>2020-06-29</date><risdate>2020</risdate><abstract>We develop a refined singularity analysis for the Ricci flow by investigating
curvature blow-up rates locally. We first introduce general definitions of Type
I and Type II singular points and show that these are indeed the only possible
types of singular points. In particular, near any singular point the Riemannian
curvature tensor has to blow up at least at a Type I rate, generalising a
result of Enders, Topping and the first author that relied on a global Type I
assumption. We also prove analogous results for the Ricci tensor, as well as a
localised version of Sesum's result, namely that the Ricci curvature must blow
up near every singular point of a Ricci flow, again at least at a Type I rate.
Finally, we show some applications of the theory to Ricci flows with bounded
scalar curvature.</abstract><doi>10.48550/arxiv.2006.16227</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Analysis of PDEs Mathematics - Differential Geometry |
| title | A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature |
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