{"title":"Convergence of curve shortening flow to translating soliton","authors":"Beomjun Choi, K. Choi, P. Daskalopoulos","doi":"10.1353/ajm.2021.0027","DOIUrl":null,"url":null,"abstract":"abstract:This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\\Bbb{R}^2$ under the $\\alpha$-curve shortening flow for exponents $\\alpha>{1\\over 2}$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $\\alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $\\alpha=1$, and we prove for all exponents up to the critical case $\\alpha>{1\\over 2}$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"143 1","pages":"1043 - 1077"},"PeriodicalIF":1.7000,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2021.0027","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"American Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1353/ajm.2021.0027","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: 2
Abstract
abstract:This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\Bbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha>{1\over 2}$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $\alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $\alpha=1$, and we prove for all exponents up to the critical case $\alpha>{1\over 2}$.
期刊介绍:
The oldest mathematics journal in the Western Hemisphere in continuous publication, the American Journal of Mathematics ranks as one of the most respected and celebrated journals in its field. Published since 1878, the Journal has earned its reputation by presenting pioneering mathematical papers. It does not specialize, but instead publishes articles of broad appeal covering the major areas of contemporary mathematics. The American Journal of Mathematics is used as a basic reference work in academic libraries, both in the United States and abroad.