关于空间形式中三调和超曲面的正规稳定性。

IF 1.2 2区数学 Q1 MATHEMATICS

Journal of Geometric Analysis Pub Date : 2023-01-01 Epub Date: 2023-08-29 DOI:10.1007/s12220-023-01414-7

Volker Branding

{"title":"关于空间形式中三调和超曲面的正规稳定性。","authors":"Volker Branding","doi":"10.1007/s12220-023-01414-7","DOIUrl":null,"url":null,"abstract":"This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere <math><mrow><mi>ϕ</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mi>m</mi></msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math> is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"33 11","pages":"355"},"PeriodicalIF":1.2000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10465648/pdf/","citationCount":"0","resultStr":"{\"title\":\"On the Normal Stability of Triharmonic Hypersurfaces in Space Forms.\",\"authors\":\"Volker Branding\",\"doi\":\"10.1007/s12220-023-01414-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere <math><mrow><mi>ϕ</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mi>m</mi></msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math> is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.\",\"PeriodicalId\":56121,\"journal\":{\"name\":\"Journal of Geometric Analysis\",\"volume\":\"33 11\",\"pages\":\"355\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10465648/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Geometric Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s12220-023-01414-7\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2023/8/29 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometric Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s12220-023-01414-7","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/8/29 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

本文研究三调和映射的稳定性，特别是三调和超曲面的稳定性。在导出了关于三调和映射稳定性的一些一般性陈述之后，我们关注空间形式中三调和超曲面的稳定性，其中我们特别注意它们的法向稳定性。我们证明了欧氏空间中常平均曲率的三调和超曲面相对于正态变化是弱稳定的，而双曲空间中常均值曲率的三谐超曲面对于正态变化则是稳定的。对于球形目标的情况，我们证明了小的本征三谐超球面的法向指数ξ:Sm（1/3）↪Sm+1等于1，并对真三调和Clifford环面的正规稳定性作了一些评论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On the Normal Stability of Triharmonic Hypersurfaces in Space Forms.

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere $ϕ : S^{m} (1 / \sqrt{3}) ↪ S^{m + 1}$ is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Geometric Analysis 数学-数学

CiteScore

2.00

自引率

9.10%

发文量

290

审稿时长

3 months

期刊介绍： JGA publishes both research and high-level expository papers in geometric analysis and its applications. There are no restrictions on page length.