{"title":"渐近双曲流形上的规定非正标量曲率与利希诺维奇方程的应用","authors":"Romain Gicquaud","doi":"10.4310/cag.2023.v31.n8.a6","DOIUrl":null,"url":null,"abstract":"We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following [$\\href{https://dx.doi.org/10.4310/CAG.2018.v26.n5.a5}{14}$, $\\href{https://doi.org/10.1090/S0002-9947-1995-1321588-5}{26}$], we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Prescribed non-positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation\",\"authors\":\"Romain Gicquaud\",\"doi\":\"10.4310/cag.2023.v31.n8.a6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following [$\\\\href{https://dx.doi.org/10.4310/CAG.2018.v26.n5.a5}{14}$, $\\\\href{https://doi.org/10.1090/S0002-9947-1995-1321588-5}{26}$], we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.\",\"PeriodicalId\":50662,\"journal\":{\"name\":\"Communications in Analysis and Geometry\",\"volume\":\"27 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-08-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Analysis and Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/cag.2023.v31.n8.a6\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cag.2023.v31.n8.a6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Prescribed non-positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following [$\href{https://dx.doi.org/10.4310/CAG.2018.v26.n5.a5}{14}$, $\href{https://doi.org/10.1090/S0002-9947-1995-1321588-5}{26}$], we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
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