Inégalités de hardy sur les variétés riemanniennes non-compactes

The Yamabe problem on stratified spaces

“…As an alternate path to proving this we could use the argument in [10], which uses integration by parts and hence requires only the density of functions with compact support in the smooth locus.…”

“…It is not hard to find examples of spaces (Z, k), even with just isolated conic singularities, where −L n k is not even semi-bounded, which simply amounts to the fact that c(n) Scal k diverges to −∞ like −c/r 2 with leading coefficient larger c than the permissible Hardy estimate bound (f − 1) 2 /4. This question is closely related to the problems studied in [20], see also [10] and [2] One further point which requires explanation is that in using condition 2), we use a conformal factor w which has leading coefficient along Σ equal to the eigenfunction w 0 for −L n k corresponding to the eigenvalue c(n)f (f −1). In order to stay with the class of iterated edge metrics, it is necessary that w 0 be bounded above and strictly positive, and this may fail.…”

Stability of the Cheng–Yau gradient estimate

“…For versions of Hardy's inequality in manifolds, see [Carron 1997;Ancona 1990]. In many situations the first item implies the second, as we now see:…”

Large time behaviour of heat kernels on non-compact manifolds : fast and slow decays

“…This example was later improved in [2]. Again, in view of Theorem 2.3 below, this means that the heat diffusion on a manifold can stay relatively fast, even though the isoperimetric profile is relatively bad.…”

“…First, Carron noticed in [2], [3] that the estimate y^{v) < Cv 110 is equivalent for D > 2 to the Sobolev inequality 11/11 ^ ^qiv/i^v/eWM), and that the latter implies The heat flow cannot decrease quickly unless it has room to escape! However, in general, i.e.…”