论文标题
磁盘区域在平均凸Riemannian $ n $ manifolds中最小化
Disks area-minimizing in mean convex Riemannian $n$-manifolds
论文作者
论文摘要
我们证明了不平等的有效性，涉及该面积的平均值和沉浸式磁盘边界的长度，其边界在定向的紧凑型歧管中是同型非平凡的曲线，该曲线具有凸平的曲率边界，正质量曲率，并在$ \ mathbb {D}^2^2^2 \ tires time t^\ time tires time t^2^2^2^^^^with and consem corvation carvation carvation car。 $ \ mathbb {d}^2 $是一个磁盘，$ t^n $是$ n $维圆环。当边界完全是大地测量时，我们还证明了平等情况的刚性结果。这可以看作是由于\ cite {amb}中卢卡斯·安布罗齐奥（LucasAmbrózio）到更高维度而导致的结果的部分概括。
We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in \cite{AMB} to higher dimensions.
