We establish an analogue of Ratner’s orbit closure theorem for an action of any connected subgroup generated by unipotent elements in SO(n,1) in the space Gamma\SO(n,1), assuming that the core of the associated hyperbolic manifold M=Gamma\H^d is compact with totally geodesic boundary. For dimension 3, this was proved earlier by McMullen-Mohammadi-Oh. In a higher dimensional case, the possibility of accummulation on closed orbits of intermediate groups causes serious new issues, but in the end, all orbit closures of unipotent flows are homogeneous in the correct sense. Our result implies that
1. the closure of any horocycle in M is a properly immersed submanifold
2. the closure of any geodesic plane intersecting the core of M is a properly immersed submanifold;
3. any infinite sequence of maximal properly immersed geodesic planes intersecting the core of M becomes dense in M.
(This talk is based on joint work with Minju Lee).