Minimal and Constant mean curvature surfaces in $\mathbb S^3$ foliated by circles
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We classify minimal surfaces in $\mathbb S^3$ which are foliated by circles and ruled constant mean curvature (cmc) surfaces in $\mathbb S^3$. First we show that minimal surfaces in $\mathbb S^3$ which are foliated by circles are either ruled (that is, foliated by geodesics) or rotationally symmetric (that is, invariant under an isometric $\mathbb S^1$-action which fixes a geodesic). Secondly, we show that, locally, there is only one ruled cmc surface in $\mathbb S^3$ up to isometry for each nonnegative mean curvature. We give a parametrization of the ruled cmc surface in $\mathbb S^3$ (cf. Theorem 3).
Abstract
We classify minimal surfaces in $\mathbb S^3$ which are foliated by circles and ruled constant mean curvature (cmc) surfaces in $\mathbb S^3$. First we show that minimal surfaces in $\mathbb S^3$ which are foliated by circles are either ruled (that is, foliated by geodesics) or rotationally symmetric (that is, invariant under an isometric $\mathbb S^1$-action which fixes a geodesic). Secondly, we show that, locally, there is only one ruled cmc surface in $\mathbb S^3$ up to isometry for each nonnegative mean curvature. We give a parametrization of the ruled cmc surface in $\mathbb S^3$ (cf. Theorem 3).
참고문헌 (5)
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