论文标题
完全弯曲的3个manifolds中完全测量的叶子和最小表面的变形
Deformations of Totally Geodesic Foliations and Minimal Surfaces in Negatively Curved 3-Manifolds
论文作者
论文摘要
让$ g_t $成为一个平滑的1参数家族，该家族是从双曲线度量开始的封闭双曲线3个manifold $ m $上的负弯曲度量的家族。我们构建了Grassmann Bundle $ gr_2（m）切线的2套叶子的叶子，其叶子是（升降）最小表面，$（m，g_t）$。这些叶子是$ gr_2（m）$的叶面的变形（升降机）完全从通用盖$ \ mathbb {h}^3 $向下投射下来的地理飞机。只要叶子（投影至$ m $）的主要曲率的平方之和在正常方向上比环境RICCI曲率的幅度小，我们的施工继续工作。在本文的第二部分中，我们提供了一些应用程序，并构建了$ gr_2（m）$无法承认上述叶面的负弯曲指标。
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to $M$) of the leaves remains pointwise smaller in magnitude than the ambient Ricci curvature in the normal direction. In the second part of the paper we give some applications and construct negatively curved metrics for which $Gr_2(M)$ cannot admit a foliation as above.
