Abstract
We prove that finite perimeter subsets of \(\mathbb {R}^{n+1}\) with small isoperimetric deficit have n-dimensional boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori \(L_p\)-norm mean curvature bounds for \(p\in [n-1,n]\). As an application, we answer a question raised by B. Colbois concerning the almost extremal hypersurfaces for Chavel’s inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Mongraphs. The Clarendon Press, Oxford University Press, New York (2000)
Aubry, E., Grosjean, J.-F.: Spectrum of hypersurfaces with small extrinsic radius or large \(\lambda _1\) in Euclidean spaces. J. Funct. Anal. 271(5), 1213–1242 (2016)
Aubry, E., Grosjean, J.-F.: Metric shape of hypersurfaces with small extrinsic radius or large \(\lambda _1\) (2012). arXiv:1210.5689(preprint)
Bernstein, F.: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60, 117–136 (1905)
Bonnesen, T.: Über die isoperimetrische Defizite ebener Figuren. Math. Ann. 91, 252–268 (1924)
Carron, G.: Stabilité isopérimétrique. Math. Ann. 306, 323–340 (1996)
Chavel, I.: On a Hurwitz’ method in isoperimetric inequalities. Proc. Am. Math. Soc. 71(2), 275–279 (1978)
Cicalese, M., Leonardi, G.P.: A selection principle for the sharp quantitative isoperimetric inequality. Arch. Rat. Mech. Anal. 206(2), 617–643 (2012)
Duggan, J.P.: \(W^{2, p}\) regularity for varifolds with mean curvature Comm. Partial Differ. Equations 11(9), 903–926 (1986)
Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)
Fuglede, B.: Stability in the isoperimetric problem for convex or nearly spherical domains in \(\mathbb{R}^n\). Trans. Am. Math. Soc. 314, 619–638 (1989)
Fuglede, B.: Bonnesen’s inequality for isoperimetric deficiency of closed curves in the plane. Geom. Dedic. 38(3), 283–300 (1991)
Fusco, N., Gelli, M.S., Pisante, G.: On a Bonnesen type inequality involving the spherical deviation. J. Math. Pures Appl. (9) 98(6), 616–632 (2012)
Fusco, N., Julin, V.: A strong form of the quantitative isoperimetric inequality (2011). arXiv:1111.4866v1
Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168(3), 941–980 (2008)
Gromov, M., Pansu, P., Lafontaine, J.: Structures métriques pour les variétés riemanniennes, Cedic/Fernand Nathan (1979)
Hall, R.R.: A quantitative isoperimetric inequality in n-dimensional space. J. Reine Angew. Math. 428, 161–176 (1992)
Hall, R.R., Hayman, W.K., Weitsman, A.: On asymmetry and capacity. J. Anal. Math. 56, 87–123 (1991)
Simon, L.: Lectures on Geometric Measure Theory. In: Proceedings of the Centre for Mathematical Analysis, v. 3. Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Aubry, E., Grosjean, JF. On the boundary of almost isoperimetric domains. Math. Z. 297, 399–427 (2021). https://doi.org/10.1007/s00209-020-02515-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-020-02515-7