×

Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov. (On an isoperimetric inequality which generalizes that of Paul Lévy-Gromov). (French) Zbl 0571.53027

In this paper we improve the Paul Levy-Gromov isoperimetric inequality. If we define h(\(\beta)\) for \(0\leq \beta \leq 1\) to be the infimum of vol(\(\partial \Omega)\) for domains \(\Omega\) such that \(vol(\Omega)=\beta vol(M)\) on a compact manifold M, we compare h(\(\beta)\) to the same function (which is known) on a sphere of radius R depending on the quantities d and \(r_{\min}\times d^ 2\) (where \(r_{\min}\) is the infimum of the lowest eigenvalue of the Ricci curvature on M and d its diameter). Some applications to lower bounds and pinching results for the first eigenvalue of the Laplacian are given as well as upper estimates of the heat kernel.

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [AN] Almgren, F.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc.4, 165 (1976) · Zbl 0327.49043
[2] [B-G] Berard, P., Gallot, S.: Inégalités isopérimétriques pour l’équation de la chaleur et applications à l’estimation de quelques invariants géométriques. Séminaire Goulaouic-Meyer-Schwartz, 1983-1984. Ecole Polytechnique, Palaiseau
[3] [B-M] Berard, P., Meyer, D.: Inégalités isopérimétriques et applications. Ann. Scient. Ec. Norm. Sup.15, 513-542 (1982)
[4] [CE] Croke, C.B.: An eigenvalues pinching theorem. Invent. math.68, 253-256 (1982) · Zbl 0505.53018 · doi:10.1007/BF01394058
[5] [G-S] Grove, K., Shiohama, K.: A generalized sphere theorem. Ann. Math.106, 201-211 (1977) · Zbl 0357.53027 · doi:10.2307/1971164
[6] [GT] Gallot, S.: Inégalités isopérimétriques, courbure de Ricci et invariants géométriques II. Note C.R. Acad. Sc. Paris296, 365-368 (1983)
[7] [GV 1] Gromov, M.: Paul Levy’s isoperimetric inequality. Prépublication I.H.E.S. (1980)
[8] [GV 2] Gromov, M.: Structures métriques pour les variétés riemanniennes. Rédigé par J. Lafontaine et P. Pansu, Cedic-Nathan (1981)
[9] [H-K] Heintze, E., Karcher, H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. Ec. Norm. Super.11, 451-470 (1978) · Zbl 0416.53027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.