Tubular Surfaces Whose Gauss Map N Satisfies $\Delta^{II}N = \Lambda N$

TL;DR

This paper characterizes certain tubular surfaces in Euclidean 3-space whose Gauss map satisfies a specific Laplace equation, showing that only circular cylinders meet these criteria.

Contribution

It proves that circular cylinders are the only tubular surfaces with a Gauss map of coordinate finite II-type satisfying the Laplace relation.

Findings

Circular cylinders are the unique solutions.

Gauss map satisfies the Laplace equation with respect to second fundamental form.

Characterization of tubular surfaces with finite II-type Gauss map.

Abstract

In this paper, we consider tubes in the Euclidean 3-space whose Gauss map N is of coordinate finite II-type, i.e., the position vector N satisfies the relation $\Delta^{II}N = \Lambda N$, where $\Delta^{II}$ is the Laplace operator with respect to the second fundamental form I of the surface and $\Lambda$ is a square matrix of order 3. We show that circular cylinders are the only class of surfaces mentioned above of coordinate finite I-type Gauss map.