A sharp quantitative version of Alexandrov's theorem via the method of moving planes

By Giulio Ciraolo, Luigi Vezzoni. 2015

48 citations recorded by the source corpus.

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 $r_e -r_i \leq C \, osc(H)$, 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 of this theorem, 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c64c6e474518d89745fdab24a05d46d80f50df5e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

On the Shape of Compact Hypersurfaces with Almost‐Constant Mean Curvature (2015, background): “More precisely, denoting by A the second fundamental form of @, in [9] it is proved the existence of c.”
Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature (2015, methodology): “Ωc dx dy |x− y|n+2 s , Ω = R \ Ω , corresponding to some value of s ∈ (0, 1/2) (see for instance [18, 2, 25, 6, 22, 20, 21, 19]).”
Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities (2017, methodology): “lts derived by the authors in [MP]. More importantly, for the cases N= 2,3 but with a weaker deviation and for a more general class of hypersurfaces, gains the (optimal) Lipschitz stability proved in [CV] and [KM, Theorem 1.8] for strictly mean convex hypersurfaces, by using the uniform deviation. A”