Sufficient conditions for constant mean curvature surfaces to be round

We study under what condition a constant mean curvature surface can be round: i) If the boundary of a compact immersed disk type constant mean curvature surface in R-3 consists of lines of curvature and has less than 4 vertices with angle < pi, then the surface is spherical; ii) A compact immersed disk type capillary surface with less than 4 vertices in a domain of R3 bounded by spheres or planes is spherical; iii) The mean curvature vector of a compact embedded capillary hypersurface of R-n with smooth boundary in an unbounded polyhedral domain with unbalanced boundary should point inward; iv) If the kth order (2 less than or equal to k less than or equal to n - 1) mean curvature of a compact immersed constant mean curvature hypersurface of R-n without boundary is constant, then the hypersurface is a sphere.