Properties and Transformations of Weingarten Surfaces

The Weingarten relations satisfied by rotationally symmetric surfaces in Euclidean 3-space E3 are considered from three points of view: restrictions on the slope of the relation at umbilic points, the action of SL2® as fractional linear transformations on the space of curvatures, and variational formulations for the relations. With regard to the first, we obtain bounds on the slope of a Weingarten relation in terms of the fall off of the radii of curvature at an umbilic point. This generalizes recent work by a number of authors. For the second, we show that the action descends from curvature space to E3 and splits into three natural geometric actions. This is applied to a class of Weingarten surfaces, called semi-quadratic, on which the action is shown to be transitive. Finally, a natural Lagrangian formulation is given for certain types of Weingarten relations and stability established.