On the Invariant Theory of Weingarten Surfaces in Euclidean Space

TL;DR

This paper establishes that strongly regular Weingarten surfaces in Euclidean space can be locally parametrized by geometric principal parameters and are uniquely determined by their structural functions and a differential equation.

Contribution

It proves the existence of geometric principal parameters for strongly regular Weingarten surfaces and characterizes them via structural functions and a differential equation.

Findings

Strongly regular Weingarten surfaces admit local geometric principal parameters.

Such surfaces are uniquely determined by structural functions and a normal curvature differential equation.

Applications include characterizations of minimal, constant mean curvature, and constant Gauss curvature surfaces.

Abstract

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature.