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| 摘要: |
| 本文研究了S2+p中2维子流形的莫比乌斯刚性问题. 设M2是2 + p维单位球S2+p中的无脐子流形, M2在S2+p的莫比乌斯变换群下的四个莫比乌斯基本量为莫比乌斯度量g, Blaschke张量A, 莫比乌斯形式Φ 以及莫比乌斯第二基本形式B, 利用不等式估计, 证明了下列刚性定理: 设x : M2 → S2+p是2 + p维单位球S2+p中莫比乌斯形式消失的2维紧致子流形, Blaschke张量A的行列式Det A = c(const) > 0, 若tr \begin{document}$A \ge \frac{1}{4}$\end{document}, 那么x(M2) 莫比乌斯等价于S2+p中常曲率极小子流形或者\begin{document}${S^3}\left( {\frac{1}{{\sqrt {1 + {c^2}} }}} \right)$\end{document}中环面\begin{document}${S^1}\left( r \right) \times {S^1}\left( {\sqrt {\frac{1}{{1 + {c^2}}} - {r^2}} } \right)$\end{document}, 其中\begin{document}${r^2} = \frac{{2 - \sqrt {1 - 64c} }}{{4 + \left( {1 + {c^2}} \right)}}$\end{document}. 本文的证明补充了文献[3] 中2维子流形情形. |
| 关键词: 2维子流形 莫比乌斯度量 莫比乌斯形式 莫比乌斯第二基本形式 Blaschke张量 |
| DOI： |
| 分类号:O186.12 |
| 基金项目: |
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| STUDY ON 2-DIMENSIONAL SUBMANIFOLDS WITH CONSTANT DETERMINANT OF BLASCHKE TENSOR |
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Ying-jia YU,Zhen GUO
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| Abstract: |
| In this paper, we study the rigidity of 2-dimensional submanifolds in S2+p. Let M2 be a 2-dimensional submanifold in the (2+p)-dimensional unit sphere S2+p without umbilic points. Four basic invariants of M2 under the Moebius transformation group of S2+p are Moebius metric g, Blaschke tensor A, Moebius form Φ and Moebius second fundamental form B. In this paper, by using inequality estimation, we proved the following rigidity theorem: Let x : M2 → S2+p be a 2-dimensional compact submanifold in the (2 + p)-dimensional unit sphere S2+p with vanishing Moebius form Φ and Det A = c(const) > 0, if tr \begin{document}$A \ge \frac{1}{4}$\end{document}, then either x(M2) is Moebius equivalent to a minimal submanifold with constant scalar curvature in S2+p, or \begin{document}${S^1}\left( r \right) \times {S^1}\left( {\sqrt {\frac{1}{{1 + {c^2}}} - {r^2}} } \right)$\end{document} in \begin{document}${S^3}\left( {\frac{1}{{\sqrt {1 + {c^2}} }}} \right)$\end{document}; where \begin{document}${r^2} = \frac{{2 - \sqrt {1 - 64c} }}{{4 + \left( {1 + {c^2}} \right)}}$\end{document}. Our results complement the case 2-dimensional submanifolds in document[3]. |
| Key words: 2-dimensional submanifolds Moebius metric Moebius form Moebius second fundamental form Blaschke tensor |
