Properties and Transformations of Weingarten Surfaces

TL;DR

This paper explores the properties, transformations, and variational formulations of rotationally symmetric Weingarten surfaces in Euclidean 3-space, providing bounds, symmetry actions, and stability results for these geometric relations.

Contribution

It introduces bounds on the slope of Weingarten relations at umbilic points, analyzes the action of SL2(R) on curvature space, and establishes a Lagrangian framework with stability results for specific Weingarten surfaces.

Findings

Bounds on the slope of Weingarten relations at umbilic points.

Decomposition of SL2(R) action into three geometric actions.

A Lagrangian formulation and stability analysis for certain Weingarten relations.

Abstract

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(R) 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.