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.

令\ (g_t \)是一个平滑的 1 参数族负曲线度量，在闭合双曲 3-流形M 上从双曲度量开始。我们构造了切线 2 平面的 Grassmann 丛\ (Gr_2 (M) \)的叶面，其叶子是\ ((M, g_t) \) 中的最小表面（的升力）。这些叶理是\ (Gr_2 (M) \)的叶理变形（升力）从通用覆盖\ ({\ mathbb {H}} ^ 3 \)向下投影的完全测地平面。只要（投影到M) 的叶子在数量上仍然比法线方向的环境 Ricci 曲率小。在论文的第二部分，我们给出了一些应用并构建了负曲线度量，其中\ (Gr_2 (M) \)不能承认上述的叶理。