The isoperimetric profile of homogeneous Riemannian manifolds.

*(English)*Zbl 1035.53069After a large and very interesting introduction related to the isoperimetric profile of a complete Riemannian manifold, the author concludes with a general question connected to the isoperimetric profile of a connected complete Riemannian manifold \(X\) such that the isometry group \(G\) of \(X\) acts quasi-transitively, that is, such that the orbit space \(G/X\) is compact (\(G\) may be a non-connected Lie group) (a so-called homogeneous Riemannian manifold).

So, the isoperimetric profile of a (non-compact) homogeneous Riemannian manifold is computed up to a multiplicative constant.

This computation is realized by constructing the exhaustions of a such manifold by Følner sets which give estimates for the distribution of the volume.

The main result of the paper (Theorem 2.1) proves the existence of only three very different isoperimetric profiles and the fact that the isoperimetric profile governs the asymptotic of the heat kernel decay on the diagonal and vice-versa.

Particularly, the isoperimetric profiles of finitely generated discrete subgroups of Lie groups are computed by discretization.

So, the isoperimetric profile of a (non-compact) homogeneous Riemannian manifold is computed up to a multiplicative constant.

This computation is realized by constructing the exhaustions of a such manifold by Følner sets which give estimates for the distribution of the volume.

The main result of the paper (Theorem 2.1) proves the existence of only three very different isoperimetric profiles and the fact that the isoperimetric profile governs the asymptotic of the heat kernel decay on the diagonal and vice-versa.

Particularly, the isoperimetric profiles of finitely generated discrete subgroups of Lie groups are computed by discretization.

Reviewer: Liliana Răileanu (Iaşi)