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On the shape of compact hypersurfaces with almost-constant mean curvature. (English) Zbl 1368.53004

For a connected bounded open set \(\Omega\subset\mathbb R^{n+1}\), \(n\geq 2\), with \(C^2\) boundary, we set \[ H_0=\frac{nP(\Omega)}{(n+1)|\Omega|}, \] where \(P(\Omega)\) and \(|\Omega|\) are the distributional perimeter of \(\Omega\) and the Lebesgue measure of \(\Omega\), respectively. Then the Alexandrov deficit of \(\Omega\) is defined by \[ \delta(\Omega)=\frac{\| H-H_0\|_{C^0(\partial\Omega)}}{H_0}, \] where \(H\) is the scalar mean curvature of \(\partial\Omega\) with respect to the outer unit normal to \(\Omega\).
Combining a mix of different ideas from elliptic PDE theory, global geometric identities, and geometric measure theory, the authors describe the shape of \(\Omega\) with small \(\delta(\Omega)\), and prove that the distance of an almost-constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of \(H\).

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
76D45 Capillarity (surface tension) for incompressible viscous fluids
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