It is shown that a minimal surface in ℍ2×ℝ is invariant under a one-parameter group of screw motions if and only if it lies in the associate family of helicoids. It is also shown that the conjugate surfaces of the parabolic and hyperbolic helicoids in ℍ2×ℝ are certain types of catenoids.

It is shown that a spacelike maximal surface in the three dimensional Lorentz-Minkowski space can be extended analytically if it meets a spacelike plane at a constant hyperbolic angle.

Let f be a smooth nonconstant function defined on an n dimensional ball and zero on the boundary n − 1 dimensional sphere. It is shown that the graph of f is a spherical cap (1) if f is positive and if the ratio Hk/Hr is a nonzero constant for 1 ≤ k < r ≤ n on the graph of f or (2) if the ratio Hn/Hk is a nonzero constant on the graph of f.

It is shown that an immersion of n dimensional compact oriented manifold without boundary into the n+1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if one of the mean curvature function H_l does not vanish and the ratio H_k/H_l is constant, 1\leq k, l \leq n, k\ne l.

It is wellknown that a compact embedded hypersurface of the Euclidean space withoutboundary is a round sphere if one of mean curvature functions is constant. Inthis note, we show that a compact embedded hypersurface of the Euclidean space(and other constant curvature spaces) without boundary is a round sphere if theratio of some two mean curvature functions isconstant.

1991 Mathematics Subject Classification 53C40, 53C20.

The following Bernstein-type theorem in hyperbolic spaces is proved. Let ∑ be a non-zero constant mean curvature complete hypersurface in the hyperbolic space ℍn. Suppose that there exists a one-to-one orthogonal projection from ∑ into a horosphere. (1) If the projection is surjective, then ∑ is a horosphere. (2) If the projection is not surjective and its image is simply connected, then ∑ is a hypersphere.

We solve the question raised by Barbosa and do Carmo as to whether there exists a complete, noncompact stably immersed surface in R3 with nonzero constant mean curvature. We show that such a surface is necessarily minimal, that is, its mean curvature is zero.