Ruled minimal surfaces in the three-dimensional Heisenberg group

Abstract

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.

Ruled minimal surfaces in the Berger sphere

“…As in our previous studies [1,5] which dealt with ruled minimal surfaces in S 2 × R, H 2 × R and in the three dimensional Heisenberg group Nil 3 , the crucial step in the proof is to show that the ruling geodesics must be either horizontal or vertical. A vector field or a geodesic is called horizontal if it is orthogonal to the Hopf fibers everywhere and is called vertical if it is tangent to the Hopf fibers everywhere (see Proposition 2.1).…”

“…If there are any differences in the proof from that of [5], they are the introduction of the new orthonormal frame (X t , V, W ) in §4 defined on the ruled surfaces and the introduction of homothetic metric. In our earlier studies, computations were manageable without using this new frame, however we think in the present case the introduction of this new frame just makes the computations manageable.…”

On the geometry of non-degenerate surfaces in Lorentzian homogeneous $3$-manifolds

“…Thus, there are defined two different mean curvature functions in any non-degenerate surface in Nil 3 1 : H R and H L respectively. In this context, Shin, Kim, Koh, Lee and Yang [13] classified non-degenerate surfaces in Nil 3 1 such that H R = H L = 0 as open pieces of a horizontal plane, a vertical plane, a helicoid or a hyperboloid paraboloid.…”

On the geometry of some solvable extensions of the Heisenberg group

“…Among Heisenberg groups, the three-dimensional Heisenberg group H 3 has attracted a special attention of geometers. For example, in the Riemannian case we refer to [19], [4] and in the Lorentzian case we refer to [17], [3]. Also recently the existence of major differences on the three-dimensional Heisenberg group in Riemannian and Lorentzian cases has been shown in [18].…”