Codimension two mean curvature flow of entire graphs

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-10-10 DOI:10.1112/jlms.13000

Andreas Savas Halilaj, Knut Smoczyk

{"title":"Codimension two mean curvature flow of entire graphs","authors":"Andreas Savas Halilaj, Knut Smoczyk","doi":"10.1112/jlms.13000","DOIUrl":null,"url":null,"abstract":"We consider the graphical mean curvature flow of maps <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <msup>\n <mi>R</mi>\n <mi>m</mi>\n </msup>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n </mrow>\n <annotation>$\\mathbf {f}:{\\mathbb {R}^{m}}\\rightarrow {\\mathbb {R}^{n}}$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$m\\geqslant 2$</annotation>\n </semantics></math>, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [Ann. of Math. (2) 130:3(1989), 453–471]. In the case of uniformly area decreasing maps <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <msup>\n <mi>R</mi>\n <mi>m</mi>\n </msup>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$\\mathbf {f}:{\\mathbb {R}^{m}} \\rightarrow {\\mathbb {R}^{2}}$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$m\\geqslant 2$</annotation>\n </semantics></math>, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 5","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.13000","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.13000","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We consider the graphical mean curvature flow of maps $f : R^{m} \to R^{n}$ , $m ⩾ 2$ , and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [Ann. of Math. (2) 130:3(1989), 453–471]. In the case of uniformly area decreasing maps $f : R^{m} \to R^{2}$ , $m ⩾ 2$ , we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.

查看原文本刊更多论文

整图的二维平均曲率流

我们考虑映射 f : R m → R n 的图形平均曲率流 $\mathbf {f}：{\mathbb {R}^{m}}\rightarrow {\mathbb {R}^{n}}$ , m ⩾ 2 $m\geqslant 2$, 并基于适当沉浸子曼形体的新版最大值原理，推导出演化图的增长率估计值，该原理扩展了 Ecker 和 Huisken 在其开创性论文 [Ann.(2) 130:3(1989), 453-471]。在均匀面积递减映射 f : R m → R 2 $\mathbf {f}:{\mathbb {R}^{m}} 的情况下。\rightarrow {\mathbb {R}^{2}}$ , m ⩾ 2 $m\geqslant 2$，我们利用这个最大原则来证明图形性和面积递减属性是保留的。此外，如果初始图形在无穷远处渐近圆锥形，我们证明归一化平均曲率流平滑地收敛于自扩展流。

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

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.