论文标题
在$ s^{n+1} $的嵌入式最小超曲面上
On embedded minimal hypersurfaces in $S^{n+1}$ with symmetries
论文作者
论文摘要
在本说明中，我们概括了由于ROS引起的Clifford圆环的表征。令$ f：m \ rightarrow s^{n+1} $为嵌入式封闭的最小超曲面。假设有$（n+2）$ s^{n+1} $彼此垂直的$ s^{n+1} $的大hyperspheres，因此相对于它们，$ m $是对称的。令$ s $表示$ f $的第二个基本形式的长度的平方，让$ \ bar s = \ frac {1} {vol（m）} \ int_ {m {m {m} sd m $是$ s $的平均值。然后$ \ bar s \ geq n $具有平等持有时，只有$ f $是clifford torus $ c_ {m，n-m} $。可以将其重写为Simons的类型定理：如果$ 0 \ leq \ int_m（n-s）d m $，则$ s \ equiv0 $或$ s \ equiv n $。这部分回答了Perdomo的猜想。此外，建立了$ f $的Willmore Energy的估计：$ W（M）\ GEQ N^{\ frac {n} {2}}} vol（m）$。
In this note, we generalize a characterization of the Clifford torus due to Ros. Let $f:M\rightarrow S^{n+1}$ be an embedded closed minimal hypersurface. Assume there are $(n+2)$ great hyperspheres of $S^{n+1}$ perpendicular to each other, such that $M$ is symmetric with respect to them. Let $S$ denote the square of the length of the second fundamental form of $f$ and let $\bar S=\frac{1}{Vol(M)}\int_{M} Sd M$ be the average of $S$. Then $\bar S\geq n$ with equality holding if and only if $f$ is the Clifford torus $C_{m,n-m}$. It can be rewritten as a Simons' type theorem: If $0\leq \int_M (n-S)d M$, then either $S\equiv0$ or $S\equiv n$. This answers partially a conjecture by Perdomo. Moreover, the estimate of the Willmore energy of $f$ is built: $W(M)\geq n^{\frac{n}{2}}Vol(M)$.
