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.
Xuyong JIANG
,
Hejun SUN
,
Peibiao ZHAO
. RIGIDITY RESULTS FOR SELF-SHRINKING SURFACES IN $\mathbb{R}^4$[J]. Acta mathematica scientia, Series B, 2021
, 41(5)
: 1417
-1427
.
DOI: 10.1007/s10473-021-0502-9
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