Abstract
In this paper, we consider the stability of quermassintegral inequalities along a inverse curvature flow. We choose a special rescaling of the flow such that the k-th quermassintegral is decreasing and the \(k-1\)-th quermassintegral is preserved. Along this rescaled flow, we prove that the decreasing rate of the k-th quermassintegral is faster than the Fraenkel asymmetry of the domain along the flow when approaching the sphere. This leads to the stability inequality of quermassintegral inequalities for nearly spherical sets using the flow method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cicalese, M., Leonardi, G.P.: A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206(2), 617–643 (2012)
Chang, S.-Y.A., Wang, Y.: On Aleksandrov–Fenchel inequalities for \(k\)-convex domains. Milan J. Math. 79(1), 13–38 (2011)
Chang, S.-Y.A., Wang, Y.: Inequalities for quermassintegrals on \(k\)-convex domains. Adv. Math. 248, 335–377 (2013)
Chang, S.-Y.A., Wang, Y.: Some higher order isoperimetric inequalities via the method of optimal transport. Int. Math. Res. Not. IMRN 0(24), 6619–6644 (2014)
Fusco, N., Julin, V.: A strong form of the quantitative isoperimetric inequality. Calc. Var. Partial Differ. Equ. 50(3–4), 925–937 (2014)
Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168(3), 941–980 (2008)
Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)
Fuglede, B.: Stability in the isoperimetric problem. Bull. Lond. Math. Soc. 18(6), 599–605 (1986)
Fuglede, B.: Stability in the isoperimetric problem for convex or nearly spherical domains in \({ R}^n\). Trans. Am. Math. Soc. 314(2), 619–638 (1989)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)
Guan, P., Li, J.: The quermassintegral inequalities for \(k\)-convex starshaped domains. Adv. Math. 221(5), 1725–1732 (2009)
Guan, P., Li, J.: Isoperimetric type inequalities and hypersurface flows. J. Math. Study 54, 56–80 (2021)
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.W.: On asymmetry and capacity. J. Analyse Math. 56, 87–123 (1991)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)
Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co. Inc., River Edge (1996)
Osserman, R.: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86(1), 1–29 (1979)
Urbas, J.I.E.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(3), 355–372 (1990)
VanBlargan, C., Wang, Y.: Quantitative quermassintegral inequalities for nearly spherical sets. Preprint, arXiv:2201.04256 (2022)
Wang, Y.: Michael–Simon inequalities for \(k\)-th mean curvatures. Calc. Var. Partial Differ. Equ. 51(1–2), 117–138 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. A. Chang.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
VanBlargan, C., Wang, Y. Stability of quermassintegral inequalities along inverse curvature flows. Calc. Var. 63, 69 (2024). https://doi.org/10.1007/s00526-024-02674-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-024-02674-z