Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 5, 2016

Isoperimetric Symmetry Breaking: a Counterexample to a Generalized Form of the Log-Convex Density Conjecture

  • Frank Morgan


We give an example of a smooth surface of revolution for which all circles about the origin are strictly stable for fixed area but small isoperimetric regions are nearly round discs away from the origin.


[1] Gregory R. Chambers. Proof of the Log-Convex Density Conjecture. J. Eur. Math. Soc. (to appear) Search in Google Scholar

[2] Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, and Weitao Zhu. Balls isoperimetric inRn with volume and perimeter densities rm and rk. arxiv:1610.05830, 2016. Search in Google Scholar

[3] Max Engelstein, Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard. Isoperimetric problems on the sphere and on surfaces with density. New York J. Math., 15:97–123, 2009. Search in Google Scholar

[4] Pengfei Guan, Junfang Li, and Mu-Tao Wang. A volume preserving flow and the isoperimetric problem in warped product spaces. arxiv:1609.08238, 2016. Search in Google Scholar

[5] Sean Howe. The log-convex density conjecture and vertical surface area in warped products. Adv. Geom., 15(4):455–468, 2015. 10.1515/advgeom-2015-0026Search in Google Scholar

[6] Frank Morgan. The log-convex density conjecture. Frank Morgan’s blog. Search in Google Scholar

[7] Frank Morgan, Michael Hutchings, and Hugh Howards. The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature. Trans. Amer. Math. Soc., 352(11):4889–4909, 2000. 10.1090/S0002-9947-00-02482-XSearch in Google Scholar

[8] Frank Morgan and David L. Johnson. Some sharp isoperimetric theorems for Riemannian manifolds. Indiana Univ. Math. J., 49(3):1017–1041, 2000. 10.1512/iumj.2000.49.1929Search in Google Scholar

[9] Manuel Ritoré. Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces. Comm. Anal. Geom., 9(5):1093–1138, 2001. 10.4310/CAG.2001.v9.n5.a5Search in Google Scholar

Received: 2016-10-12
Accepted: 2016-11-16
Published Online: 2016-12-5

© 2016 Frank Morgan

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.3.2023 from
Scroll Up Arrow