论文标题
具有正交边界的曲率varifolds
Curvature Varifolds with Orthogonal Boundary
论文作者
论文摘要
我们考虑$ \barΩ\ subset {\ mathbb r}^n $类别的$ s^m_ \ perp（ω）$ $ m $ - 二维表面，该{\ mathbb r}^n $与边界沿边界上相交的$ s = \ partialω$正交。 $ s^m_ \ perp（ω）$中的一块仿射$ m $ - 平面称为正交切片。我们证明了该区域的估算值，在三种情况下，第二个基本形式的$ l^p $ - 构成：首先，当$ω$没有正交切片时，如果所有正交切片都是拓扑磁盘，则$ m = p = 2 $第二，如果表面限制在$ω$的所有$ω$的社区中，则首先是$ m = 2 $。正交性约束对于曲率varifolds的公式较弱。我们对消失的曲率进行了分类。作为一个应用程序，我们证明了任何$ω$的存在正交$ 2 $ -Varifold的存在，该$ 2 $ -VARIFOLD最小化了整数可纠正类中的$ l^2 $曲率。
We consider the class $S^m_\perp(Ω)$ of $m$-dimensional surfaces in $\barΩ \subset {\mathbb R}^n$ which intersect $S = \partial Ω$ orthogonally along the boundary. A piece of an affine $m$-plane in $S^m_\perp(Ω)$ is called an orthogonal slice. We prove estimates for the area by the $L^p$-integral of the second fundamental form in three cases: first when $Ω$ admits no orthogonal slices, second for $m = p = 2$ if all orthogonal slices are topological disks, and finally for all $Ω$ if the surfaces are confined to a neighborhood of $S$. The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any $Ω$ the existence of an orthogonal $2$-varifold which minimizes the $L^2$ curvature in the integer rectifiable class.
