an:03861962
Zbl 0542.53025
B??rard, Pierre; Gallot, Sylvestre
In??galit??s isop??rim??triques pour l'??quation de la chaleur et application ?? l'estimation de quelques invariants. (Isoperimetric inequalities for the heat ??quation and application to the estimation of some invariants)
FR
S??min. Goulaouic-Meyer-Schwartz 1983-1984, ??quat. d??riv. part., Expos?? No. 15, 34 p. (1984).
1984
j
53C20 53C65 58J50
heat kernel; geometric isoperimetric inequalities; Ricci curvature; eigenvalues of the Laplacian
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.
R.Sperb