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Analytic inequalities, and rough isometries between non-compact Riemannian manifolds. (English) Zbl 0593.53026
Curvature and topology of Riemannian manifolds, Proc. 17th Int. Taniguchi Symp., Katata/Jap. 1985, Lect. Notes Math. 1201, 122-137 (1986).
[For the entire collection see Zbl 0583.00022.]
In this paper, the author continues his previous works on rough isometries between non-compact Riemannian manifolds [J. Math. Soc. Japan 37, 391-413 (1985; Zbl 0554.53030); ibid. 38, 227-238 (1986; Zbl 0577.53031)]. For a complete Riemannian manifold X, its Sobolev constant \(S_{\ell,m}(X)\) is introduced by \[ S_{\ell,m}(X)=\inf_{u\in C_ 0^{(\infty)}(X)}(\| \nabla u\|_{L^{\ell}(X)}/=u\|_ Lm/(m-1)_{(X)}), \] and the main reslt of this paper claims that whether \(S_{\ell,m}(X)>0\) or not is preserved by rough isometries under a certain uniformness condition posed on Riemannian manifolds. The proof is done by a kind of difference approximation methods. Using this method, the author also discusses the relation between the work of Kesten on random walks on discrete groups, and the inequalities of Cheeger-Buser which give estimates of the ”PoincarĂ© constant” \(S_{2,2}(X)\) in terms of an isoperimetric constant.

53C20 Global Riemannian geometry, including pinching