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.{{cite encyclopedia|first1=Alexander Danilovich|last1=Alexandrov|title=Uniqueness theorem for surfaces in the large|series=2|volume=21|encyclopedia=American Mathematical Society Translations|publisher=American Mathematical Soc.|date=1962|pages=412–416}}{{cite journal|first1=Alexander Danilovich|last1=Alexandrov|title=A characteristic property of spheres|journal=Annali di Matematica |volume=58|pages=303–315|date=1962|doi=10.1007/BF02413056}} 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 $\Omega\subset \mathbb{R}^n$ 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.{{cite journal|first1=Rolando|last1=Magnanini|first2=Giorgio|last2=Poggesi|date=2017|title=Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities|volume=69|journal=Indiana University Mathematics Journal|doi=10.1512/iumj.2020.69.7925|arxiv=1708.07392}}{{cite arXiv|eprint=1811.05202|first1=Giulio|last1=Ciraolo|first2=Alberto|last2=Roncoroni|title=The method of moving planes: a quantitative approach|date=2018|page=1}}

Literature

{{cite arXiv|eprint=1811.05202|first1=Giulio|last1=Ciraolo|first2=Alberto|last2=Roncoroni|title=The method of moving planes: a quantitative approach|date=2018|page=1}}
{{cite encyclopedia|title=Nine Papers on Topology, Lie Groups, and Differential Equations|volume=21|encyclopedia=American Mathematical Society Translations|series=2|publisher=American Mathematical Soc.|first1=Yurii Mikhailovich|last1=Smirnov|first2=Alexander Danilovich|last2=Aleksandrov|date=1962|isbn=0821817213}}

References

Category:Theorems in differential geometry