Minimal surfaces and genuine laminations

This dissertation covers two different research topics.; The first one is a study on minimal planes in hyperbolic space by using geometrical analysis methods. We show a generic finiteness result for least area planes in H3. Moreover, we prove that the space of minimal immersions of disk into H3 is a submanifold of product bundle over a space of immersions of circle into $S2\infty$(H3) and the bundle projection map is when restricted to this submanifold is Fredholm of index zero. Using this, we also show that the space of minimal planes with smooth boundary curve at infinity is a manifold.; The second one is concerning geometric topology, and a study to show existence of special structures on a class of 3-manifolds. We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are indeed the “straightening” of the coarse laminations defined by Calegari, by using minimal surface techniques. Moreover, when you collapse these laminations, you can get a topological pseudo-Anosov flow as defined by Mosher.