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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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참고문헌 (6)
참고문헌 신청
A. D. Alexandrov / 1962 / Uniqueness theorems for surfaces in the large V / Amer. Math. Soc. Transl 21 : 412 ~ 416
D. Gilbarg / 2001 / Classics in mathematics / Springer-Verlag
H. Hopf / 1989 / Differential Geometry in the Large / Springer-Verlag
J. McCuan / 1997 / Symmetry via spherical re ection and spanning drops in a wedge / Pacific J. Math 180 (2) : 291 ~ 323
J. McCuan / 2000 / Symmetry via spherical re ection / J. of Geom. Analysis 10 (3) : 545 ~ 564
D. Gilbarg / 2001 / Classics in mathematics / Springer-Verlag
H. Hopf / 1989 / Differential Geometry in the Large / Springer-Verlag
J. McCuan / 1997 / Symmetry via spherical re ection and spanning drops in a wedge / Pacific J. Math 180 (2) : 291 ~ 323
J. McCuan / 2000 / Symmetry via spherical re ection / J. of Geom. Analysis 10 (3) : 545 ~ 564
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