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.

在本文中，我们在关于高斯图像的一些假设下，给出了正确浸入ℝ ^{4 中的}完全自收缩表面的一些刚性结果。更准确地说，我们证明这必须是一个平面，前提是高斯图投影的图像位于开放的半球或\({{\mathbb{S}}^2}(1/\sqrt 2 )\backslash \bar {\mathbb{S}}_ + ^1(1/\sqrt 2 )\)。我们还给出了正确浸入ℝ ^{4 中}的完全自收缩表面的分类，前提是高斯图投影的图像位于一些封闭的半球。作为上述结果的应用，我们对 Zhou 的结果给出了新的证明。此外，我们建立了伯恩斯坦型定理。