Dimensions at infinity for Riemannian manifolds. (English) Zbl 0847.53022

From the introduction: “This paper is essentially a survey, with however some variations or new points of view on already published results. The main theme is the study of the relationship between various Sobolev type inequalities on manifolds. In the first part, we introduce a scale of dimensions at infinity adapted to manifolds of polynomial growth, in which we recast the results of [the author, J. Funct. Anal. 136, No. 1, 81-113 (1996); D. Bakry, M. Ledoux, L. Coste-Saloff and the author, Sobolev inequalities in disguise (preprint); and M. Ledoux and the author, Ark. Mat. 32, No. 1, 63-77 (1994; Zbl 0826.53035)]. In the second one, we show how Poincaré inequalities allow at the same time to go down in the scale and to refine it into a scale of global inequalities. In the third part, we take up the more general situation where the volume growth of the manifolds is not necessarily governed by a power function”.


53C20 Global Riemannian geometry, including pinching


Zbl 0826.53035
Full Text: DOI


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