Symmetry about circles and constant mean curvature surface
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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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References
[1] A. D. Alexandrov, Uniqueness theorems for surfaces in the large V, Amer. Math. Soc. Transl. 21 (1962), 412–416. Google Scholar
[2] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Classics in mathematics, Springer-Verlag, (2001). Google Scholar
[3] H. Hopf, Differential Geometry in the Large, Lect. Notes Math 1000, Springer Verlag, Berlin (1989). Google Scholar
[4] J. McCuan, Symmetry via spherical reflection and spanning drops in a wedge, Pacific J. Math. 180 (2) (1997), 291–323. Google Scholar
[5] J. McCuan, Symmetry via spherical reflection, J. of Geom. Analysis, Vol. 10, Issue 3 (2000), 545–564. Google Scholar
[6] 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