{"title":"翘积流形中具有规定韦氏曲率的 k 凸超曲面","authors":"","doi":"10.1016/j.na.2024.113640","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider Weingarten curvature equations for <span><math><mi>k</mi></math></span>-convex hypersurfaces with <span><math><mrow><mi>n</mi><mo><</mo><mn>2</mn><mi>k</mi></mrow></math></span> in a warped product manifold <span><math><mrow><mover><mrow><mi>M</mi></mrow><mo>¯</mo></mover><mo>=</mo><mi>I</mi><msub><mrow><mo>×</mo></mrow><mrow><mi>λ</mi></mrow></msub><mi>M</mi></mrow></math></span>. Based on the conjecture proposed by Ren–Wang in Ren and Wang (2020), which is valid for <span><math><mrow><mi>k</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span>, we derive curvature estimates for equation <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>ν</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface <span><math><mi>Σ</mi></math></span> satisfying the above equation by the degree theory under some sufficient conditions.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"k-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds\",\"authors\":\"\",\"doi\":\"10.1016/j.na.2024.113640\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we consider Weingarten curvature equations for <span><math><mi>k</mi></math></span>-convex hypersurfaces with <span><math><mrow><mi>n</mi><mo><</mo><mn>2</mn><mi>k</mi></mrow></math></span> in a warped product manifold <span><math><mrow><mover><mrow><mi>M</mi></mrow><mo>¯</mo></mover><mo>=</mo><mi>I</mi><msub><mrow><mo>×</mo></mrow><mrow><mi>λ</mi></mrow></msub><mi>M</mi></mrow></math></span>. Based on the conjecture proposed by Ren–Wang in Ren and Wang (2020), which is valid for <span><math><mrow><mi>k</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span>, we derive curvature estimates for equation <span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>ν</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface <span><math><mi>Σ</mi></math></span> satisfying the above equation by the degree theory under some sufficient conditions.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001597\",\"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":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001597","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文考虑在翘曲积流形M¯=I×λM中n<2k的k凸超曲面的韦氏曲率方程。基于任旺在 Ren and Wang (2020) 中提出的对 k≥n-2 有效的猜想,我们通过直接证明得出了方程 σk(κ)=ψ(V,ν(V)) 的曲率估计。此外,我们还通过度理论在一些充分条件下得到了满足上述方程的星形紧凑超曲面 Σ 的存在性结果。
k-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds
In this paper, we consider Weingarten curvature equations for -convex hypersurfaces with in a warped product manifold . Based on the conjecture proposed by Ren–Wang in Ren and Wang (2020), which is valid for , we derive curvature estimates for equation through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface satisfying the above equation by the degree theory under some sufficient conditions.
期刊介绍:
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