\(\mathbb{S}^5\)中嘉当极小超曲面的曲率特征
A curvature characterization of the Cartan minimal hypersurface in $\mathbb S^5$
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中文总结 AI 辅助
研究\(\mathbb{S}^5\)中满足\(|W|^2 = 2|\mathring{\text{Ric}}|^2\)（欧拉平衡条件）的闭极小超曲面，通过相关理论证明此类超曲面要么全测地，要么与嘉当极小超曲面全等。
中文摘要 AI 辅助
劳森证明了\(\mathbb{S}^5\)中一个非全测地的爱因斯坦极小超曲面与克利福德超曲面\(\mathbb{S}^2(1/\sqrt{2})\times\mathbb{S}^2(1/\sqrt{2})\)全等。通过嘉当和大月薰的工作可知，\(\mathbb{S}^5\)中一个非全测地的局部共形平坦极小超曲面是大月薰型的，包括克利福德超曲面\(\mathbb{S}^1(1/2)\times\mathbb{S}^3(\sqrt{3}/2)\)。本文研究\(\mathbb{S}^5\)中满足\(|W|^2 = 2|\mathring{\text{Ric}}|^2\)的闭极小超曲面\(M\)（欧拉平衡条件），证明这样的超曲面要么是全测地的，要么与嘉当极小超曲面全等。
英文摘要
Lawson showed that a non-totally geodesic Einstein minimal hypersurface in $\mathbb S^5$ is congruent to the Clifford hypersurface $\mathbb S^2(1/\sqrt2)\times \mathbb S^2(1/\sqrt2).$ It is also known, by work of Cartan and Ôtsuki, that a non-totally geodesic locally conformally flat minimal hypersurface in $\mathbb S^5$ is of Ôtsuki type, including the Clifford hypersurface $\mathbb S^1(1/2)\times \mathbb S^3(\sqrt3/2).$ In this paper we study closed minimal hypersurfaces $M$ in $\mathbb S^5$ satisfying $|W|^2=2|\mathring{\operatorname{Ric}}|^2,$ where $W$ is the Weyl tensor and $\mathring{\operatorname{Ric}}$ is the trace-free Ricci tensor. We call this the Euler-balanced condition. We prove that such a hypersurface is either totally geodesic or congruent to the Cartan minimal hypersurface.
