For n ≥ 2 we define a notion of umbilicity for hypersurfaces in the Heisenberg group Hn. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant p(or horizontal)-mean curvature in Hn up to Heisenberg translations.

Abstract. Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n + 1. Let M be the universal covering of M. Let Φ denote a CR developing mapwhere S 2n+1 is the standard unit sphere in complex n + 1-space C n+1 . Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n ≥ 3. In the case n = 2, we also show that Φ is injective under the condition: s(M ) < 1. It then follows that M is uniformizable.

Abstract. We study immersed, connected, umbilic hypersurfaces in the Heisenberg group Hn with n ≥ 2. We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.