Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov. (On an isoperimetric inequality which generalizes that of Paul Lévy-Gromov) (Q1062019)

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.