A simple analytic proof of an inequality by P. Buser

Ledoux, M.

doi:10.1090/S0002-9939-1994-1186991-X

A simple analytic proof of an inequality by P. Buser
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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

Paul Lévy’s isoperimetric inequality

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