Alexandrov's soap bubble theorem explained

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.

Soap bubble theorem

Let

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

Literature

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

References

1. Encyclopedia: Alexander Danilovich. Alexandrov. Uniqueness theorem for surfaces in the large. 2. 21. American Mathematical Society Translations. American Mathematical Soc.. 1962. 412–416.
2. Alexander Danilovich. Alexandrov. A characteristic property of spheres. Annali di Matematica . 58. 303–315. 1962. 10.1007/BF02413056.
3. Rolando. Magnanini. Giorgio. Poggesi. 2017. Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities. 69. Indiana University Mathematics Journal. 10.1512/iumj.2020.69.7925. 1708.07392.
4. 1811.05202. Giulio. Ciraolo. Alberto. Roncoroni. The method of moving planes: a quantitative approach. 2018. 1.