Circle-foliated Minimal surfaces in $4$-dimensional space forms
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1433-1443
https://doi.org/10.4134/BKMS.2015.52.5.1433
Printed September 30, 2015
Sung-Ho Park
Hankuk University of Foreign Studies
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 $\zeta \in \mathbb C \setminus \{0\}$ is also foliated by circles. In this paper, we show that every circle-foliated minimal surface in $\mathbb R^n$ 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 $H^4$.
Keywords : circle-foliated surface, minimal surface in $\mathbb S^4$ and $\mathbb H^4$
MSC numbers : 53A10, 53C12
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