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.

Soap bubble theorem

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

Literature

• Ciraolo, Giulio; Roncoroni, Alberto (2018). "The method of moving planes: a quantitative approach". arXiv:1811.05202 [math.AP].
• 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. {{cite encyclopedia}}: ISBN / Date incompatibility (help)