Lie minimal Weingarten surfaces

Abstract

We consider Lie minimal surfaces, the critical points of the simplest Lie sphere invariant energy, in Riemannian space forms. These surfaces can be characterized via their Euler-Lagrange equations, which take the form of differential equations in terms of the principal curvatures. Surfaces with constant mean curvature that satisfy these equations turn out to be rotational in their space form. We generalize in flat ambient space: here surfaces whose principal curvatures satisfy an affine relationship as well as elliptic linear Weingarten surfaces are rotational as well.