ABSTRACT. In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in H 3 . Hence, this gives a counterexample to CalabiYau conjecture for embedded minimal surfaces in negative curvature case.

We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting a transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are the straightening of the coarse laminations defined by Calegari, by using minimal surface techniques. Moreover, when we collapse these laminations, we get a topological pseudo-Anosov flow, as defined by Mosher.

We study the number of solutions of the asymptotic Plateau problem in ‫ވ‬ 3 . By using the analytical results in our previous paper, and some topological arguments, we show that there exists an open dense subset of C 3 Jordan curves in S 2 1 ‫ވ.‬ 3 / such that any curve in this set bounds a unique least area plane in ‫ވ‬ 3 .

In this paper we show a generic finiteness result for least area planes in H 3 . Moreover, we prove that the space of minimal immersions of disk into H 3 is a submanifold of product bundle over a space of immersions of circle into S 2 ∞ (H 3 ) 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.

We study the constant mean curvature (CMC) hypersurfaces in H n+1 whose asymptotic boundaries are closed codimension-1 submanifolds in S n ∞ (H n+1 ). We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area-minimizing, convex hull property, exchange roundoff trick to CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space. (2000). 53A10. Mathematics Subject Classification