A characterization of hyperbolic caps in the steady state space

We obtain two characterizations of planar discs and hyperbolic caps of the steady state space in the family of compact spacelike surfaces with constant mean curvature: (1) they are the only ones spanning a circle; (2) they are the only capillary surfaces with boundary in a slice. We also obtain some results of symmetry when the boundary is a simple closed curve contained in a slice.