Abstract
It is our purpose to study complete self-shrinkers in Euclidean space. By making use of the generalized maximum principle for \(\mathcal {L}\)-operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in \(\mathbb R^3\). Ding and Xin (Trans Am Math Soc 366:5067–5085, 2014) have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.
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Authors would like to thank Professor Wei Guoxin for fruitful discussions.
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Dedicated to Professor Yoshihiko Suyama for his 70th birthday.
Qing-Ming Cheng: Research partially Supported by JSPS Grant-in-Aid for Scientific Research (B) No. 24340013 and Challenging Exploratory Research No. 25610016.
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Cheng, QM., Ogata, S. 2-Dimensional complete self-shrinkers in \(\mathbf {R}^3\) . Math. Z. 284, 537–542 (2016). https://doi.org/10.1007/s00209-016-1665-2
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DOI: https://doi.org/10.1007/s00209-016-1665-2