Laguerre isothermic surfaces

Abstract

Surfaces with isothermic spherical representation of their lines of curvature, called Laguerre isothermic surfaces, are examined. The first chapter gives an introduction to Laguerre geometry. In the second chapter it is shown how a second-order linear nonhomogeneous differential equation is associated with Laguerre isothermic surfaces. Position vectors of a family of parallel Laguerre isothermic surfaces may be expressed in terms of solutions of this equation. Furthermore, different types of transformations of Laguerre isothermic surfaces are discussed. First it is shown that the action of the group SL(2,C) on solutions of the above mentioned equation produces Laguerre transformations of the associated surfaces. It is also proved that a Baecklund transformation of a Laguerre isothermic surface corresponds to a Darboux transformation of the potential of the associated equation. Using two Baecklund transformations of a Laguerre isothermic surface a new Laguerre isothermic surface may be constructed. Moreover, a Darboux transformation of Laguerre isothermic surfaces is introduced and it is shown that the resulting surfaces are again Laguerre isothermic. In the last chapter a Weierstrass-type representation of Laguerre isothermic Laguerre minimal surfaces is given.