Alexandrov’s soap bubble theorem
From HandWiki
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 [math]\displaystyle{ \Omega\subset \mathbb{R}^n }[/math] be a bounded connected domain with a boundary [math]\displaystyle{ \Gamma=\partial\Omega }[/math] that is of class [math]\displaystyle{ C^2 }[/math] with a constant mean curvature, then [math]\displaystyle{ \Gamma }[/math] is a sphere.[3][4]
Literature
- 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. 21. American Mathematical Soc.. ISBN 0821817213.
References
- ↑ Alexandrov, Alexander Danilovich (1962). "Uniqueness theorem for surfaces in the large". American Mathematical Society Translations. 2. 21. American Mathematical Soc.. pp. 412–416.
- ↑ Alexandrov, Alexander Danilovich (1962). "A characteristic property of spheres". Annali di Matematica 58: 303–315. doi:10.1007/BF02413056.
- ↑ Magnanini, Rolando; Poggesi, Giorgio (2017). "Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities". Indiana University Mathematics Journal 69. doi:10.1512/iumj.2020.69.7925.
- ↑ Ciraolo, Giulio; Roncoroni, Alberto (2018). "The method of moving planes: a quantitative approach". p. 1. arXiv:1811.05202.
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