## 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
Full Text:

### References:

 [1] Allard, On the first variation of a varifold, Ann. of Math. (2) 95 pp 417– (1972) · Zbl 0252.49028 [2] Ambrosetti, Nonlinear analysis and semilinear elliptic problems (2007) · Zbl 1125.47052 [3] Arnold, On the Aleksandrov-Fenchel inequality and the stability of the sphere, Monatsh. Math 115 (1-2) pp 1– (1993) · Zbl 0784.52008 [4] Brandolini, On the stability of the Serrin problem, J. Differential Equations 245 (6) pp 1566– (2008) · Zbl 1173.35019 [5] Butscher, A gluing construction for prescribed mean curvature, Pacific J. Math 249 (2) pp 257– (2011) · Zbl 1215.53013 [6] Butscher, CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (3) pp 653– (2012) · Zbl 1260.53111 [7] Caffarelli, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 pp 1457– (1994) · Zbl 0819.35016 [8] Cicalese, Improved convergence theorems for bubble clusters. I. The planar case, Indiana Univ. Math. J 65 (6) pp 1979– (2016) · Zbl 1381.49051 [9] Ciraolo, A sharp quantitative version of Alexandrov’s theorem via the method of moving planes, J. Eur. Math. Soc (2015) · Zbl 1397.53050 [10] Lellis, Zurich Lectures in Advanced Mathematics (2008) [11] De Lellis , C. Allard’s interior regularity theorem: an invitation to stationary varifolds http://user.math.uzh.ch/delellis/fileadmin/delellis/allard-24.pdf [12] Lellis, Optimal rigidity estimates for nearly umbilical surfaces, J. Differential Geom 69 (1) pp 75– (2005) · Zbl 1087.53004 [13] Lellis, A C0 estimate for nearly umbilical surfaces, Calc. Var. Partial Differential Equations 26 (3) pp 283– (2006) · Zbl 1100.53005 [14] Figalli, On the shape of liquid drops and crystals in the small mass regime, Arch. Ration. Mech. Anal 201 (1) pp 143– (2011) · Zbl 1279.76005 [15] Figalli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math 182 (1) pp 167– (2010) · Zbl 1196.49033 [16] Finn, Grundlehren der mathematischen Wissenschaften 284 (1986) [17] Fusco, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (3) pp 941– (2008) · Zbl 1187.52009 [18] Heintze, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. école Norm. Sup. (4) 11 (4) pp 451– (1978) · Zbl 0416.53027 [19] Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (2) pp 239– (1990) · Zbl 0699.53007 [20] Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom 33 (3) pp 683– (1991) · Zbl 0727.53063 [21] Kohlmann, Curvature measures and stability, J. Geom 68 (1-2) pp 142– (2000) · Zbl 0981.52003 [22] Krummel, Isoperimetry with upper mean curvature bounds and sharp stability estimates (2016) · Zbl 1368.49054 [23] Maggi, An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics (2012) · Zbl 1255.49074 [24] Montiel, Differential geometry pp 279– (1991) [25] Perez , D. On nearly umbilical surfaces. Thesis, University of Zürich 2011 http://user.math.uzh.ch/delellis/uploads/media/Daniel.pdf [26] Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J 26 (3) pp 459– (1977) · Zbl 0391.53019 [27] Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamericana 3 (3-4) pp 447– (1987) · Zbl 0673.53003 [28] Schneider, A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature, Manuscripta Math 69 (3) pp 291– (1990) · Zbl 0713.52003 [29] Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal 43 pp 304– (1971) · Zbl 0222.31007 [30] Simon, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3 (1983) [31] Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (4) pp 697– (1976) · Zbl 0341.35031 [32] Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv 83 (3) pp 539– (2008) · Zbl 1154.53007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.