Minimal surfaces inS3foliated by circles

Abstract

We study minimal surfaces in the unit sphere S 3 that 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 only two types of minimal surfaces are foliated by circles crossing the principal lines at a constant angle. The first type of surfaces are foliated by great circles that are bisectrices of the principal lines, and we show that these are the examples of Lawson. The second type, 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 and an application to the theory of minimal foliated semisymmetric hypersurfaces in R 4 .