Laguerre Geometry of Surfaces in R 3

Abstract

Let f : M → R3 be an oriented surface with non–degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L(f) = f (H2 – K)/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R3. And we give a classification theorem of surfaces in R3 with vanishing Laguerre form.