Cheeger’s constant in balls and isoperimetric inequality on Riemannian manifolds

García León, Joel

doi:10.1090/S0002-9939-08-08824-2

Cheeger’s constant in balls and isoperimetric inequality on Riemannian manifolds
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by Joel García León

Proc. Amer. Math. Soc. 136 (2008), 4445-4452

DOI: https://doi.org/10.1090/S0002-9939-08-08824-2

Published electronically: July 15, 2008

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Abstract:

We prove isoperimetric inequality on a Riemannian manifold, assuming that the Cheeger constant for balls satisfies a certain estimation.

References

Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
Fan Chung, Alexander Grigor′yan, and Shing-Tung Yau, Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs, Comm. Anal. Geom. 8 (2000), no. 5, 969–1026. MR 1846124, DOI 10.4310/CAG.2000.v8.n5.a2
Faber, C. Beweiss, das unter allen homogenen Membrane von gleicher Spannung die kreisförmige die tiefsten Grundton gibt. Sitzungsber.-Bayer. Akad. Wiss., Math.-Phys. Munich (1923), 169-172.
García León, J. Cheeger Constant and Isoperimetric Inequalities on Riemannian Manifolds, Ph.D. thesis, Imperial College, London, 2005.
Alexander Grigor′yan, Estimates of heat kernels on Riemannian manifolds, Spectral theory and geometry (Edinburgh, 1998) London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 140–225. MR 1736868, DOI 10.1017/CBO9780511566165.008
E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), no. 1, 97–100 (German). MR 1512244, DOI 10.1007/BF01208645
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
Robert Osserman, Minimal varieties, Bull. Amer. Math. Soc. 75 (1969), 1092–1120. MR 276875, DOI 10.1090/S0002-9904-1969-12357-8
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
John William Strutt Rayleigh Baron, The Theory of Sound, Dover Publications, New York, N.Y., 1945. 2d ed. MR 0016009