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Abstract. We show that some class of spacelike maximal surfaces and timelike minimal surfaces match smoothly across the singular curve of the surfaces. Singular Björling representation formulae for generalized spacelike maximal surfaces and for generalized timelike minimal surfaces play important roles in the explanation of this phenomenon.

Abstract. It is well known that a compact embedded hypersurface of the Euclidean space without boundary is a round sphere if one of mean curvature functions is constant. In this note, we show that a compact embedded hypersurface of the Euclidean space (and other constant curvature spaces) without boundary is a round sphere if the ratio of some two mean curvature functions is constant.

It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three-dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three-dimensional Heisenberg group whose mean curvature is zero with respect to both the standard Riemannian metric and the standard Lorentzian metric.

To the memory of Professor Jeong-Seon Baek. We provide two characterizations of helicoids in ‫ޓ‬ 2 ‫ޒ×‬ and in ‫ވ‬ 2 ‫.ޒ×‬ First, we show that any nontrivial ruled minimal surface in ‫ޓ‬ 2 × ‫ޒ‬ and in ‫ވ‬ 2 × ‫ޒ‬ is a part of a helicoid. Second, we also show that these surfaces can be characterized as the only surface with zero mean curvature with respect to both the Riemannian product metric and the Lorentzian product metric on ‫ޓ‬ 2 × ‫ޒ‬ or ‫ވ‬ 2 × ‫.ޒ‬ MSC2000: 53A35.