{"title":"超曲面的标量曲率幂的演化","authors":"R. Alessandroni, C. Sinestrari","doi":"10.2422/2036-2145.2010.3.05","DOIUrl":null,"url":null,"abstract":"We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"24 1","pages":"541-571"},"PeriodicalIF":1.2000,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"33","resultStr":"{\"title\":\"Evolution of hypersurfaces by powers of the scalar curvature\",\"authors\":\"R. Alessandroni, C. Sinestrari\",\"doi\":\"10.2422/2036-2145.2010.3.05\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.\",\"PeriodicalId\":50966,\"journal\":{\"name\":\"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze\",\"volume\":\"24 1\",\"pages\":\"541-571\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2010-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"33\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2422/2036-2145.2010.3.05\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2422/2036-2145.2010.3.05","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Evolution of hypersurfaces by powers of the scalar curvature
We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.
期刊介绍：
The Annals of the Normale Superiore di Pisa, Science Class, publishes papers that contribute to the development of Mathematics both from the theoretical and the applied point of view. Research papers or papers of expository type are considered for publication.
The Annals of the Normale Scuola di Pisa - Science Class is published quarterly
Soft cover, 17x24
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