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CIRCLE-FOLIATED MINIMAL SURFACES IN 4-DIMENSIONAL SPACE FORMS

  • PARK, SUNG-HO (MAJOR IN MATHEMATICS GRADUATE SCHOOL OF EDUCATION HANKUK UNIVERSITY OF FOREIGN STUDIES)
  • Received : 2014.02.10
  • Published : 2015.09.30

Abstract

Catenoid and Riemann's minimal surface are foliated by circles, that is, they are union of circles. In $\mathbb{R}^3$, there is no other nonplanar example of circle-foliated minimal surfaces. In $\mathbb{R}^4$, the graph $G_c$ of w = c/z for real constant c and ${\zeta}{\in}\mathbb{C}{\backslash}\{0}$ is also foliated by circles. In this paper, we show that every circle-foliated minimal surface in $\mathbb{R}$ is either a catenoid or Riemann's minimal surface in some 3-dimensional Affine subspace 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-foliated minimal surfaces in $S^4$ and $H^4$.

Acknowledgement

Supported by : Hankuk University of Foreign Studies Research Fund

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