Inégalités isopérimétriques, courbure de Ricci et invariants géométriques. I. (French) Zbl 0535.53034

Gallot, Sylvestre

C. R. Acad. Sci., Paris, Sér. I 296, 333-336 (1983).

Summary: We give a sharp estimate for some isoperimetric constants on Riemannian manifolds. As consequences, we obtain: a lower bound (as universal as possible) of the first eigenvalue of the Laplace-Beltrami operator for the Dirichlet problem on a domain of a Riemannian manifold, an improvement of Cheeger’s inequality \(\lambda_ 1>h^ 2/4\), an explicit and sharp calculus of the constants involved in the inequalities linked to Sobolev’s inclusions. This last result will enable us to estimate several topological and Riemannian invariants (see the paper reviewed below).

MSC:

53C20	Global Riemannian geometry, including pinching
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

isoperimetric constants; lower bound; first eigenvalue; Laplace-Beltrami operator; Dirichlet problem; Cheeger’s inequality; Sobolev’s inclusions