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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Contracting convex immersed closed plane curves with slow speed of curvature
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by Yu-Chu Lin, Chi-Cheung Poon and Dong-Ho Tsai PDF
Trans. Amer. Math. Soc. 364 (2012), 5735-5763 Request permission

Abstract:

The authors study the contraction of a convex immersed plane curve with speed $\frac {1}{\alpha }k^{\alpha }$, where $\alpha \in (0,1]$ is a constant, and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. They also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when $\alpha =1$), this translational self-similar solution is the familiar “Grim Reaper” (a terminology due to M. Grayson).
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Additional Information
  • Yu-Chu Lin
  • Affiliation: Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
  • MR Author ID: 843221
  • Email: yclin@math.ncku.edu.tw
  • Chi-Cheung Poon
  • Affiliation: Department of Mathematics, National Chung Cheng University, Chiayi 621, Taiwan
  • Email: ccpoon@math.ccu.edu.tw
  • Dong-Ho Tsai
  • Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
  • Email: dhtsai@math.nthu.edu.tw
  • Received by editor(s): October 12, 2010
  • Published electronically: June 20, 2012
  • Additional Notes: The third author’s research was supported by the NCTS and the NSC of Taiwan under grant number 96-2115-M-007-010-MY3.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 5735-5763
  • MSC (2010): Primary 53C44, 35K15, 35K55
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05611-X
  • MathSciNet review: 2946930