Gioacchino Antonelli, Marco Pozzetta, Daniele Semola
{"title":"非负Ricci曲率非紧流形中大等周集的平均唯一性","authors":"Gioacchino Antonelli, Marco Pozzetta, Daniele Semola","doi":"10.1002/cpa.22252","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>M</mi>\n <mi>n</mi>\n </msup>\n <mo>,</mo>\n <mi>g</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(M^n,g)$</annotation>\n </semantics></math> be a complete Riemannian manifold which is not isometric to <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math>, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>⊂</mo>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {G}\\subset (0,\\infty)$</annotation>\n </semantics></math> with density 1 at infinity such that for every <span></span><math>\n <semantics>\n <mrow>\n <mi>V</mi>\n <mo>∈</mo>\n <mi>G</mi>\n </mrow>\n <annotation>$V\\in \\mathcal {G}$</annotation>\n </semantics></math> there is a unique isoperimetric set of volume <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> in <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$M$</annotation>\n </semantics></math>; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>I</mi>\n <mi>n</mi>\n </msub>\n <mo>⊂</mo>\n <mrow>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$I_n\\subset (0,\\infty)$</annotation>\n </semantics></math> such that isoperimetric sets with volumes <span></span><math>\n <semantics>\n <mrow>\n <mi>V</mi>\n <mo>∈</mo>\n <msub>\n <mi>I</mi>\n <mi>n</mi>\n </msub>\n </mrow>\n <annotation>$V\\in I_n$</annotation>\n </semantics></math> exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 9","pages":"1656-1702"},"PeriodicalIF":2.7000,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22252","citationCount":"0","resultStr":"{\"title\":\"Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature\",\"authors\":\"Gioacchino Antonelli, Marco Pozzetta, Daniele Semola\",\"doi\":\"10.1002/cpa.22252\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msup>\\n <mi>M</mi>\\n <mi>n</mi>\\n </msup>\\n <mo>,</mo>\\n <mi>g</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(M^n,g)$</annotation>\\n </semantics></math> be a complete Riemannian manifold which is not isometric to <span></span><math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mi>n</mi>\\n </msup>\\n <annotation>$\\\\mathbb {R}^n$</annotation>\\n </semantics></math>, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n <mo>⊂</mo>\\n <mo>(</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mi>∞</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathcal {G}\\\\subset (0,\\\\infty)$</annotation>\\n </semantics></math> with density 1 at infinity such that for every <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>V</mi>\\n <mo>∈</mo>\\n <mi>G</mi>\\n </mrow>\\n <annotation>$V\\\\in \\\\mathcal {G}$</annotation>\\n </semantics></math> there is a unique isoperimetric set of volume <span></span><math>\\n <semantics>\\n <mi>V</mi>\\n <annotation>$V$</annotation>\\n </semantics></math> in <span></span><math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$M$</annotation>\\n </semantics></math>; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>I</mi>\\n <mi>n</mi>\\n </msub>\\n <mo>⊂</mo>\\n <mrow>\\n <mo>(</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mi>∞</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$I_n\\\\subset (0,\\\\infty)$</annotation>\\n </semantics></math> such that isoperimetric sets with volumes <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>V</mi>\\n <mo>∈</mo>\\n <msub>\\n <mi>I</mi>\\n <mi>n</mi>\\n </msub>\\n </mrow>\\n <annotation>$V\\\\in I_n$</annotation>\\n </semantics></math> exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.</p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":\"78 9\",\"pages\":\"1656-1702\"},\"PeriodicalIF\":2.7000,\"publicationDate\":\"2025-04-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22252\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22252\",\"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":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22252","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature
Let be a complete Riemannian manifold which is not isometric to , has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set with density 1 at infinity such that for every there is a unique isoperimetric set of volume in ; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals such that isoperimetric sets with volumes exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.