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Rigidity Results for Self-Shrinking Surfaces in ℝ4


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 or Hejun Sun.

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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).

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Key words

  • self-shrinkers
  • Gauss map
  • Bernstein-type theorem
  • rigidity

2010 MR Subject Classification

  • 53C24
  • 53C42