Deformations of Totally Geodesic Foliations and Minimal Surfaces in Negatively Curved 3-Manifolds

Let $$g_t$$ be a smooth 1-parameter family of negatively curved metrics on closed hyperbolic 3-manifold M starting at the metric. We construct foliations Grassmann bundle $$Gr_2(M)$$ tangent 2-planes whose leaves are (lifts of) minimal surfaces in $$(M,g_t)$$ . These deformations foliation by totally geodesic planes projected down from universal cover $${\mathbb {H}}^3$$ Our construction continues to work as long sum squares principal curvatures (projections M) remains pointwise smaller magnitude than ambient Ricci curvature normal direction. In second part paper we give some applications and for which cannot admit above.