{"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}
引用次数: 0
Abstract
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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