A simple analytic proof of an inequality by P. Buser
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- by M. Ledoux PDF
- Proc. Amer. Math. Soc. 121 (1994), 951-959 Request permission
Abstract:
We present a simple analytic proof of the inequality of P. Buser showing the equivalence of the first eigenvalue of a compact Riemannian manifold without boundary and Cheeger’s isoperimetric constant under a lower bound on the Ricci curvature. Our tools are the Li-Yau inequality and ideas of Varopoulos in his functional approach to isoperimetric inequalities and heat kernel estimates on groups and manifolds. The method is easily modified to yield a logarithmic isoperimetric inequality involving the hypercontractivity constant of the manifold.References
- P. Bérard, G. Besson, and S. Gallot, Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov, Invent. Math. 80 (1985), no. 2, 295–308 (French). MR 788412, DOI 10.1007/BF01388608
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
- Thierry Coulhon, Sobolev inequalities on graphs and on manifolds, Harmonic analysis and discrete potential theory (Frascati, 1991) Plenum, New York, 1992, pp. 207–214. MR 1222459
- E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
- Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. MR 997938
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 2nd ed., Universitext, Springer-Verlag, Berlin, 1990. MR 1083149, DOI 10.1007/978-3-642-97242-3 M. Gromov, Paul Lévy’s isoperimetric inequality, Inst. Hautes Etudes Sci., preprint, 1980.
- Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
- Michel Ledoux, Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math. 118 (1994), no. 6, 485–510. MR 1309086
- Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
- O. S. Rothaus, Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities, J. Functional Analysis 42 (1981), no. 1, 102–109. MR 620581, DOI 10.1016/0022-1236(81)90049-5
- O. S. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal. 64 (1985), no. 2, 296–313. MR 812396, DOI 10.1016/0022-1236(85)90079-5
- N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240–260. MR 803094, DOI 10.1016/0022-1236(85)90087-4
- N. Th. Varopoulos, Small time Gaussian estimates of heat diffusion kernels. I. The semigroup technique, Bull. Sci. Math. 113 (1989), no. 3, 253–277. MR 1016211
- Shing Tung Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487–507. MR 397619
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 951-959
- MSC: Primary 53C21; Secondary 58G25
- DOI: https://doi.org/10.1090/S0002-9939-1994-1186991-X
- MathSciNet review: 1186991