On Surfaces that are Intrinsically Surfaces of Revolution

TL;DR

This paper characterizes a class of surfaces in Euclidean space with rotationally invariant fundamental forms and curvatures, proving they are necessarily of constant mean curvature and classifying special cases.

Contribution

It introduces a new class of surfaces with rotationally invariant properties and proves they must have constant mean curvature, extending the understanding of surfaces of revolution.

Findings

Surfaces with specified invariance are necessarily of constant mean curvature.

The rotational speed of principal curvature directions is constant.

The minimal case within this class is fully classified.

Abstract

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.