Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (5): 1417-1427.doi: 10.1007/s10473-021-0502-9

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RIGIDITY RESULTS FOR SELF-SHRINKING SURFACES IN $\mathbb{R}^4$

Xuyong JIANG1, Hejun SUN2, Peibiao ZHAO2   

  1. 1. Department of Mathematics, Changzhou University, Changzhou 213164, China;
    2. College of Science, Nanjing University of Science and Technology, Nanjing 210094, China
  • Received:2020-05-07 Revised:2021-05-14 Online:2021-10-25 Published:2021-10-21
  • Contact: Hejun SUN E-mail:hejunsun@163.com
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (11001130, 11871275) and the Fundamental Research Funds for the Central Universities (30917011335).

Abstract: In this paper, we give some rigidity results for complete self-shrinking surfaces properly immersed in $\mathbb{R}^4$ under some assumptions regarding their Gauss images. More precisely, we prove that this has to be a plane, provided that the images of either Gauss map projection lies in an open hemisphere or $\mathbb{S}^2(1/\sqrt{2})\backslash \bar{\mathbb{S}}^1_+(1/\sqrt{2})$. We also give the classification of complete self-shrinking surfaces properly immersed in $\mathbb{R}^4$ provided that the images of Gauss map projection lies in some closed hemispheres. As an application of the above results, we give a new proof for the result of Zhou. Moreover, we establish a Bernstein-type theorem.

Key words: self-shrinkers, Gauss map, Bernstein-type theorem, rigidity

CLC Number: 

  • 53C24
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