Gianmarco Caldini , Andrea Marchese , Andrea Merlo , Simone Steinbrüchel
{"title":"高原问题的一般唯一性","authors":"Gianmarco Caldini , Andrea Marchese , Andrea Merlo , Simone Steinbrüchel","doi":"10.1016/j.matpur.2023.10.010","DOIUrl":null,"url":null,"abstract":"<div><p>Given a complete Riemannian manifold <span><math><mi>M</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which is a Lipschitz neighborhood retract of dimension <span><math><mi>m</mi><mo>+</mo><mi>n</mi></math></span>, of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>,</mo><mi>β</mi></mrow></msup></math></span> and an oriented, closed submanifold <span><math><mi>Γ</mi><mo>⊂</mo><mi>M</mi></math></span> of dimension <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span>, which is a boundary in integral homology, we construct a complete metric space <span><math><mi>B</mi></math></span> of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>,</mo><mi>α</mi></mrow></msup></math></span>-perturbations of Γ inside <span><math><mi>M</mi></math></span>, with <span><math><mi>α</mi><mo><</mo><mi>β</mi></math></span>, enjoying the following property. For the typical element <span><math><mi>b</mi><mo>∈</mo><mi>B</mi></math></span>, in the sense of Baire categories, there exists a unique <em>m</em>-dimensional integral current in <span><math><mi>M</mi></math></span> which solves the corresponding Plateau problem and it has multiplicity one.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"181 ","pages":"Pages 1-21"},"PeriodicalIF":2.1000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782423001514/pdfft?md5=3ea9206021feeebc508e67ce882752fd&pid=1-s2.0-S0021782423001514-main.pdf","citationCount":"1","resultStr":"{\"title\":\"Generic uniqueness for the Plateau problem\",\"authors\":\"Gianmarco Caldini , Andrea Marchese , Andrea Merlo , Simone Steinbrüchel\",\"doi\":\"10.1016/j.matpur.2023.10.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given a complete Riemannian manifold <span><math><mi>M</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which is a Lipschitz neighborhood retract of dimension <span><math><mi>m</mi><mo>+</mo><mi>n</mi></math></span>, of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>,</mo><mi>β</mi></mrow></msup></math></span> and an oriented, closed submanifold <span><math><mi>Γ</mi><mo>⊂</mo><mi>M</mi></math></span> of dimension <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span>, which is a boundary in integral homology, we construct a complete metric space <span><math><mi>B</mi></math></span> of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>h</mi><mo>,</mo><mi>α</mi></mrow></msup></math></span>-perturbations of Γ inside <span><math><mi>M</mi></math></span>, with <span><math><mi>α</mi><mo><</mo><mi>β</mi></math></span>, enjoying the following property. For the typical element <span><math><mi>b</mi><mo>∈</mo><mi>B</mi></math></span>, in the sense of Baire categories, there exists a unique <em>m</em>-dimensional integral current in <span><math><mi>M</mi></math></span> which solves the corresponding Plateau problem and it has multiplicity one.</p></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"181 \",\"pages\":\"Pages 1-21\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-11-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0021782423001514/pdfft?md5=3ea9206021feeebc508e67ce882752fd&pid=1-s2.0-S0021782423001514-main.pdf\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de Mathematiques Pures et Appliquees\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782423001514\",\"RegionNum\":1,\"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":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001514","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
摘要
莫德a complete Riemannian流形M⊂Rd - which is a李普希茨邻里retract of of class M + n维度,Ch,通过面向βand an,关闭submanifoldΓ⊂of维度M in integral homology−1,which is a号边界,we建筑B a complete规space of Ch,α-perturbations ofΓinside M, with the <β、α,他会请property。对于典型元素b∈b,在贝尔范畴的意义上,存在一个独特的M维积分流,它解决了对应的平台问题,它有一个多重性。一双(M),有人认为Γ⊂Rd),其中M是M + n维度全面品种和班级Ch、β是邻里rétract李普希茨、和Γ⊂M是sous-variété封闭和导向,即整整一个已知的边缘。正在建一个完整metrique扰动Ch,αBΓ在β、α< M,满足如下:对于任何所有权B∈通用在Baire sense有整整一个独一无二的潮流中M, m-dimensionnelle和多样性,这是解决塞浦路斯问题和相应的高原。
Given a complete Riemannian manifold which is a Lipschitz neighborhood retract of dimension , of class and an oriented, closed submanifold of dimension , which is a boundary in integral homology, we construct a complete metric space of -perturbations of Γ inside , with , enjoying the following property. For the typical element , in the sense of Baire categories, there exists a unique m-dimensional integral current in which solves the corresponding Plateau problem and it has multiplicity one.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.