论文标题
3个球体中嵌入式表面属的估计值
An estimate for the genus of embedded surfaces in the 3-sphere
论文作者
论文摘要
通过完善Heintze和Karcher \ cite {hk}的体积估计值，我们获得了$ \ Mathbb s^{3} $中表面属的尖锐捏合估计，该估计涉及其可怜的无第二基础形式的规范。更具体地说，我们表明，如果$ g $是$ 3 $ 3 $二维的可定位的Riemannian歧管$ m $的封闭式表面$σ$的属，其分段曲率下面的截面为$ 1 $，则$4π^{2} g（σ） f（| \ stackrel \ circ a |）$，其中$ \ stackrel \ circ a $是无可怜的第二个基本形式，$ f $是明确的功能。结果，封闭的可定向嵌入式最小表面$σ$，均匀边界$ \ | a \ | _ {l^3（σ）} $在任何$ k \ ge2 $的$ c^k $ topology中是紧凑的。
By refining the volume estimate of Heintze and Karcher \cite{HK}, we obtain a sharp pinching estimate for the genus of a surface in $\mathbb S^{3}$, which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if $g$ is the genus of a closed orientable surface $Σ$ in a $3$-dimensional orientable Riemannian manifold $M$ whose sectional curvature is bounded below by $1$, then $4 π^{2} g(Σ) \le 2\left(2 π^{2}-|M|\right)+\int_Σ f(|\stackrel \circ A|)$, where $ \stackrel \circ A $ is the traceless second fundamental form and $f$ is an explicit function. As a result, the space of closed orientable embedded minimal surfaces $Σ$ with uniformly bounded $\|A\|_{L^3(Σ)}$ is compact in the $C^k$ topology for any $k\ge2$.
