### 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.

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PDF#### References

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DOI: http://dx.doi.org/10.11568/kjm.2017.25.4.555

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ISSN: 1976-8605 (Print), 2288-1433 (Online)

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