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Rigidity Results for Self-Shrinking Surfaces in ℝ4

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Abstract

In this paper, we give some rigidity results for complete self-shrinking surfaces properly immersed in ℝ4 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 ℝ4 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.

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Correspondence to Xuyong Jiang  (江绪永) or Hejun Sun  (孙和军).

Additional information

This work was supported by the National Natural Science Foundation of China (11001130, 11871275) and the Fundamental Research Funds for the Central Universities (30917011335).

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Jiang, X., Sun, H. & Zhao, P. Rigidity Results for Self-Shrinking Surfaces in ℝ4. Acta Math Sci 41, 1417–1427 (2021). https://doi.org/10.1007/s10473-021-0502-9

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  • DOI: https://doi.org/10.1007/s10473-021-0502-9

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