Remarks on a Variational Problem in Laguerre Geometry

Recently there has been some renewed interest in Laguerre differential geometry [1], [2], [3]. This geometry was to a large extent developed by Blaschke and his school and a large amount of material about it can be found in [4]. Consider the set of oriented spheres in Euclidean space. If we add to this set the space of oriented planes, i.e. the spheres of infinite radii, then we obtain the space of the Moebius (conformal ) geometry. This space can be identified with the deSitter space which is a Lorentzian 4-manifold of constant curvature +1. If, on the other hand, we add the spheres of zero radii, i.e. ”point spheres” then we obtain the Laguerre space which can be identified with the four dimensional Minkowski space. Any smooth immersion of an oriented surface into 3-space has a lift to the unit sphere bundle of IE called the Legendre lift. The unit sphere bundle may also be considered as the space of null lines in Minkowski space. Thus the isometry group of Minkowski space acts on the set of Legendre surfaces. The principle aim of Laguerre geometry is to study properties of the immersion which are invariant under this action. The simplest way to do this is to define at each point p of the surface an oriented 2-sphere or point sphere Y (p) which is in some sense invariant under this action. The map Y will be called the L-Gauss map. It is analogous to the conformal Gauss map of Moebius geometry and defines a spacelike immersion of the surface into Minkowski space. The geometric invariants of this spacelike immersion are exactly the Laguerre invariants of the original surface. In particular, its area defines a Laguerre invariant functional: