Alexandrov's soap bubble theorem

Alexandrov's soap bubble theorem is a mathematical theorem from geometric analysis that characterizes a sphere through the mean curvature. The theorem was proven in 1958 by Alexander Danilovich Alexandrov.^{[1] [2]} In his proof he introduced the method of moving planes, which was used after by many mathematicians successfully in geometric analysis.

Let ${\displaystyle \Omega \subset \mathbb {R} ^{n}}${\displaystyle \Omega \subset \mathbb {R} ^{n}} be a bounded connected domain with a boundary ${\displaystyle \Gamma =\partial \Omega }${\displaystyle \Gamma =\partial \Omega } that is of class ${\displaystyle C^{2}}${\displaystyle C^{2}} with a constant mean curvature, then ${\displaystyle \Gamma }${\displaystyle \Gamma } is a sphere.^{[3] [4]}

• Ciraolo, Giulio; Roncoroni, Alberto (2018). "The method of moving planes: a quantitative approach". p. 1. arXiv:1811.05202 .
• Smirnov, Yurii Mikhailovich; Aleksandrov, Alexander Danilovich (1962). "Nine Papers on Topology, Lie Groups, and Differential Equations". American Mathematical Society Translations. 2. Vol. 21. American Mathematical Soc. ISBN 0821817213.

• Differential geometry

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