Korean J. Math.  Vol 25, No 4 (2017)  pp.555-561
DOI: https://doi.org/10.11568/kjm.2017.25.4.555

Symmetry about circles and constant mean curvature surface

Sung-ho Park


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.


cmc surface, symmetry

Subject classification

53C24, 53C12


Full Text:



A. D. Alexandrov, Uniqueness theorems for surfaces in the large V, Amer. Math. Soc. Transl. 21 (1962), 412–416. (Google Scholar)

D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Classics in mathematics, Springer-Verlag, (2001). (Google Scholar)

H. Hopf, Differential Geometry in the Large, Lect. Notes Math 1000, Springer Verlag, Berlin (1989). (Google Scholar)

J. McCuan, Symmetry via spherical reflection and spanning drops in a wedge, Pacific J. Math. 180 (2) (1997), 291–323. (Google Scholar)

J. McCuan, Symmetry via spherical reflection, J. of Geom. Analysis, Vol. 10, Issue 3 (2000), 545–564. (Google Scholar)

S. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski Space, Rocky Mountain J. Math. 32 (3) (2002), 1014–1044. (Google Scholar)


  • There are currently no refbacks.

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

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr