CIRCLE-FOLIATED MINIMAL SURFACES IN 4-DIMENSIONAL SPACE FORMS

Abstract

Abstract. Catenoid and Riemann’s minimal surface are foliated by cir-cles, that is, they are union of circles. In R 3 , there is no other nonpla-nar example of circle-foliated minimal surfaces. In R 4 , the graph G c ofw = c/z for real constant c and ζ ∈C\{0}is also foliated by circles. In thispaper, we show that every circle-foliated minimal surface in R n is eithera catenoid or Riemann’s minimal surface in some 3-dimensional Affinesubspace or a graph surface G c in some 4-dimensional Affine subspace.We use the property that G c is circle-foliated to construct circle-foliatedminimal surfaces in S 4 and H . 1. IntroductionA surface M ∈ R n is said to be circle-foliated if there is a one-parameterfamily of planes whose intersection with M are circles. The catenoid andRiemann’s minimal surface are examples of circle-foliated minimal surfaces inR 3 . Enneper proved that the planes containing the circles of a circle-foliatedminimal surface in R 3 should be parallel [2] and [7]. Then it is easy to see thatthe plane, catenoid and Riemann’s minimal surface are the only circle-foliatedminimal surfaces in R