Minimal surfaces in $S^3$ foliated by circles

TL;DR

This paper classifies minimal surfaces in the 3-sphere foliated by circles, revealing only two types: Lawson's known examples with great circles and new families with small circles, including a constructive formula.

Contribution

It proves that only two types of circle-foliated minimal surfaces exist in S^3, providing a new family with small circles and a constructive formula for these surfaces.

Findings

Only two types of circle-foliated minimal surfaces in S^3 exist.

Lawson's examples are the only known surfaces with great circles.

New minimal surfaces with small circles are constructed explicitly.

Abstract

We deal with minimal surfaces in the unit sphere $S^3$, which are one-parameter families of circles. Minimal surfaces in $\R^3$ foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in $S^3$. We prove that in $S^3$ there are only two types of minimal surfaces foliated by circles, crossing the principal lines at a constant angle. The first type surfaces are foliated by great circles, which are bisectrices of the principal lines, and we show that these minimal surfaces are the well-known examples of Lawson. The second type surfaces, which are new in the literature, are families of small circles, and the circles are principal lines. We give a constructive formula for these surfaces. An application to the theory of minimal foliated semi-symmetric hypersurfaces in $\R^4$ is given.