- journal article metadata
European Mathematical Society Publishing House
2018-02-03 23:30:01
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
20
2018
2
A sharp quantitative version of Alexandrov's theorem via the method of moving planes
Giulio
Ciraolo
Università di Palermo, Italy
Luigi
Vezzoni
Università di Torino, Italy
Alexandrov Soap Bubble Theorem, method of moving planes, stability, mean curvature, pinching
We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by osc$(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if osc$(H) \leq \varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \subset \overline{B}_{r_e} \setminus B_{r_i}$ and $r_e -r_i \leq C \, \mathrm {osc}(H)$, with $C$ depending only on $n$ and upper bounds on the surface area of $S$ and the $C^2$ regularity of $S$. Our approach is based on a quantitative study of the method of moving planes, and the quantitative estimate on $r_e-r_i$ we obtain is optimal. As a consequence, we also prove that if osc$(H)$ is small then $S$ is diffeomorphic to a sphere, and give a quantitative bound which implies that $S$ is $C^1$-close to a sphere.
Partial differential equations
Differential geometry
261
299
10.4171/JEMS/766
http://www.ems-ph.org/doi/10.4171/JEMS/766
1
31
2018