{"title":"单位球中自由边界超曲面沿反向平均曲率流的第一特征值","authors":"Pak Tung Ho , Juncheol Pyo","doi":"10.1016/j.difgeo.2023.102095","DOIUrl":null,"url":null,"abstract":"<div><p><span>In this note, we consider the first nonzero eigenvalue </span><span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> of the <em>p</em><span><span>-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse </span>mean curvature flow. We show that </span><span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> of a free boundary disk type hypersurface in the unit ball.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"First eigenvalues of free boundary hypersurfaces in the unit ball along the inverse mean curvature flow\",\"authors\":\"Pak Tung Ho , Juncheol Pyo\",\"doi\":\"10.1016/j.difgeo.2023.102095\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span>In this note, we consider the first nonzero eigenvalue </span><span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> of the <em>p</em><span><span>-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse </span>mean curvature flow. We show that </span><span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msub></math></span> of a free boundary disk type hypersurface in the unit ball.</p></div>\",\"PeriodicalId\":51010,\"journal\":{\"name\":\"Differential Geometry and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-12-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0926224523001213\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224523001213","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
First eigenvalues of free boundary hypersurfaces in the unit ball along the inverse mean curvature flow
In this note, we consider the first nonzero eigenvalue of the p-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of of a free boundary disk type hypersurface in the unit ball.
期刊介绍:
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.