Inégalités isopérimétriques pour l’équation de la chaleur et application à l’estimation de quelques invariants. (Isoperimetric inequalities for the heat équation and application to the estimation of some invariants). (French) Zbl 0542.53025

Sémin. Goulaouic-Meyer-Schwartz 1983-1984, Équat. dériv. part., Exposé No. 15, 34 p. (1984).
The authors first give an introduction, describing the various differential geometric notions, in particular curvatures. They then prove their main result giving estimates for the heat kernel on a Riemannian manifold, provided some geometric assumptions are satisfied. Their main tools are rearrangements of functions and symmetrization arguments combined with geometric isoperimetric inequalities. They also give a number of interesting geometric inequalities containing for example bounds for the Ricci curvature and the diameter of the manifold as well as inequalities for the eigenvalues of the Laplacian.
Reviewer: R.Sperb


53C20 Global Riemannian geometry, including pinching
53C65 Integral geometry
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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