Symmetry about circles and constant mean curvature surface

Abstract

We show that a closed curve invariant under inversions with respect to two intersecting circles intersecting at angle of an irrational multiple of $2\pi$ is a circle. This generalizes the well known fact that a closed curve symmetric about two lines intersecting at angle of an irrational multiple of $2\pi$ is a circle. We use the result to give a different proof of that a compact embedded cmc surface in ${\mathbb{R}}^3$ is a sphere. Finally we show that a closed embedded cmc surface which is invariant under the spherical reflections about two spheres, which intersect at an angle that is an irrational multiple of $2\pi$, is a sphere.