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Isoperimetry, decrease of the heat kernel and Riesz transformations: A counterexample. (Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: Un contre-exemple.) (French) Zbl 0826.53035

Let \(M\) be a complete, noncompact Riemannian manifold. If \(A \subset M\) is a compact subset with a regular boundary then the isoperimetric inequality \(|A|^{(N - 1)/N} \leq C |\partial A|\) implies \(\sup_{x, y \in M} p_t(x,y) \leq C' t^{-N/2}\), \(t > 0\), where \(p_t\) is a heat kernel on \(M\) [cf. N. Th. Varopoulos, C. R. Acad. Sci., Paris, Sér. I 299, 651-654 (1984; Zbl 0566.31006)]. A partial converse to the above theorem was given by N. Th. Varopoulos in [Bull. Sci. Math., II. Sér. 113, No. 3, 253-277 (1989; Zbl 0703.58052)]: if \(M\) has a bounded geometry and \(\sup_{x, y \in M} p_t (x,y) = O(t^{-N/2})\), \(t \to +\infty\) then \(|A|^{(N' - 1)/N'} \leq C|\partial A|\) if \(\dim M \leq N' \leq N/2\). In the present paper the authors show that the assumption \(N' \leq N/2\) can not be replaced by \(N' \leq N\), namely they prove:
For each integer \(N \geq 3\) and for each real number \(N' > N/2\) there exists a Riemannian manifold \(M\) of dimension \(N\) with bounded sectional curvature and with positive injectivity radius such that: i) \(\sup_{x,y \in M} p_t(x,y) = O(t^{-N/2})\), \(t \to +\infty\), where \(p_t\) is a heat kernel on \(M\). ii) The isoperimetric inequality \(|A|^{(N' - 1)/N'} \leq C |\partial A|\) does not hold for any \(C\) where \(A \subset M\) is a compact subset with regular boundary and contains a ball of fixed radius.
Reviewer: W.Mozgawa (Lublin)

MSC:

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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