Minimal surfaces in S³ foliated by circles

Nikolai Kutev and Velichka Milousheva

Vol. 248 (2010), No. 2, 335–354

DOI: 10.2140/pjm.2010.248.335

Abstract

We study minimal surfaces in the unit sphere S³ that are one-parameter families of circles. Minimal surfaces in ℝ³ foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in S³. We prove that in S³ 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 ℝ⁴.

Keywords

minimal surfaces in S³, Clifford torus, Lawson torus, harmonic map, foliated semisymmetric hypersurface, minimal hypersurface, nonlinear elliptic system

Mathematical Subject Classification 2000

Primary: 53A10, 35J60

Secondary: 53A07, 35A07

Milestones

Received: 16 July 2009

Revised: 27 November 2009

Accepted: 30 November 2009

Published: 1 December 2010