A minimal surface foliated by circles or geodesics in $d\,\mathbb{S}^3$ and $\mathbb{H}^3$

Abstract

Circles in the Minkowski space $\mathbb L^n$ are a natural generalization of circles in Euclidean space. In this paper, $d\,\mathbb S^3$ or $\mathbb H^3$ are supposed to be embedded in $\mathbb L^4$. A surface $M$ in $d\,\mathbb S^3$ or in $\mathbb H^3$ is said to be circle-foliated if $M$ is the union of a smooth one-parameter family of circles in $\mathbb L^4$. Similarly, we say that $M$ is ruled if $M$ is the union of a smooth one-parameter family of geodesics. We completely classify minimal surfaces in $d\,\mathbb S^3$ or in $\mathbb H^3$ that are circle-foliated or ruled. We show that a minimal surface in $\mathbb H^3$ foliated by circles is either ruled, rotationally symmetric or part of a pseudo-great sphere. We also show that a minimal surface in $d\,\mathbb S^3$ foliated by circles is either ruled, rotationally symmetric or part of a pseudo-great sphere. We give explicit parametrizations of all ruled minimal surfaces in $d\,\mathbb S^3$ or $\mathbb H^3$. Finally, we show that there is no ruled surface of non-zero constant mean curvature in $d\,\mathbb S^3$ or in $\mathbb H^3$.