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Ruled Laguerre minimal surfaces

By Philipp Grohs Helmut Pottmann Mikhail Skopenkov


A Laguerre minimal surface is an immersed surface in R 3 being an extremal of the functional 2 (H /K − 1)dA. In the present paper, we prove that any ruled Laguerre minimal surface distinct from a plane is up to motion a convolution of the helicoid x = y tan z, the cycloid r(t) = (t − sin t, 1 − cos t, 0) and the Plücker conoid (ax + by) 2 = z(x 2 + y 2) for some a, b ∈ R. 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 biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles i

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Year: 2010
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