49 research outputs found
Ruled Laguerre minimal surfaces
A Laguerre minimal surface is an immersed surface in the Euclidean space
being an extremal of the functional \int (H^2/K - 1) dA. In the present paper,
we prove that the only ruled Laguerre minimal surfaces are up to isometry the
surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C,
D are fixed real numbers. To achieve invariance under Laguerre transformations,
we also derive all Laguerre minimal surfaces that are enveloped by a family of
cones. The methodology is based on the isotropic model of Laguerre geometry. In
this model a Laguerre minimal surface enveloped by a family of cones
corresponds to a graph of a biharmonic function carrying a family of isotropic
circles. We classify such functions by showing that the top view of the family
of circles is a pencil.Comment: 28 pages, 9 figures. Minor correction: missed assumption (*) added to
Propositions 1-2 and Theorem 2, missed case (nested circles having nonempty
envelope) added in the proof of Pencil Theorem 4, missed proof that the arcs
cut off by the envelope are disjoint added in the proof of Lemma