Nonnegative Ricci curvature and minimal graphs with linear growth

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-08-21 DOI:10.2140/apde.2024.17.2275

Giulio Colombo, Eddygledson S. Gama, Luciano Mari, Marco Rigoli

{"title":"Nonnegative Ricci curvature and minimal graphs with linear growth","authors":"Giulio Colombo, Eddygledson S. Gama, Luciano Mari, Marco Rigoli","doi":"10.2140/apde.2024.17.2275","DOIUrl":null,"url":null,"abstract":"<p>We study minimal graphs with linear growth on complete manifolds <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow></msup></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Ric</mi><mo> ⁡ </mo>\n<mo>≥</mo> <mn>0</mn></math>. Under the further assumption that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>-th Ricci curvature in radial direction is bounded below by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi><mi>r</mi><msup><mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math>, we prove that any such graph, if nonconstant, forces tangent cones at infinity of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> to split off a line. Note that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> is not required to have Euclidean volume growth. We also show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar’s gradient estimate for minimal graphs, together with heat equation techniques. </p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/apde.2024.17.2275","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

We study minimal graphs with linear growth on complete manifolds $M^{m}$ with $Ric \geq 0$ . Under the further assumption that the $(m - 2)$ -th Ricci curvature in radial direction is bounded below by $C r {(x)}^{- 2}$ , we prove that any such graph, if nonconstant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar’s gradient estimate for minimal graphs, together with heat equation techniques.

查看原文本刊更多论文

非负里奇曲率和线性增长的最小图形

我们研究在 Ric ≥ 0 的完全流形 Mm 上具有线性增长的最小图形。在进一步假设径向的 (m-2)-th Ricci 曲率在下面以 Cr(x)-2 为界的情况下，我们证明任何这样的图形，如果是非恒定的，都会迫使 M 的无穷远处的切圆锥分裂出一条直线。请注意，M 并不需要具有欧几里得体积增长。我们还证明了 M 可能不会从任何直线上分裂出来。我们的结果与 Cheeger、Colding 和 Minicozzi 针对谐函数得到的结果相似。本文的核心是结合热方程技术，对 Korevaar 的极小图梯度估计进行了新的改进。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.