Isoperimetric estimates in low-dimensional Riemannian products

Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

“…for all smooth open submanifolds U with compact closure containing a closed disk of radius ρ, then we may take (1.17) that N = d + k, r = k. Such product manifolds have been the subject of much interest in the literature (see [37,39]). In this case the results outlined in the previous example still hold when in them we formally let β = (k − 1)/(N − 1), ν = 0; the density ρ is taken as in the previous example.…”

Some remarks on the Sobolev inequality in Riemannian manifolds

“…However, we pursue here a different class of examples, that is the one arising from product manifolds in which a factor is an Euclidean space and the other one is compact. The problem of determining the isoperimetric profile even in this specific class of Riemannian manifolds is difficult (see [29] and references therein). In our examples, if N is the topological dimension of the product manifold and k < N its dimension at infinity, one has (1.12) with λ = (N −1)/N and µ = (k−1)/k; a lengthy but elementary explicit computation yields in this case that the sup defining B 1 in (1.6) is attained as s → +∞ and…”

“…In [25] it is considered the case of the product S 1 r × R k , where S 1 r is the circle of radius r > 0; here N = k + 1 and 2 ≤ k ≤ 7; the isoperimetric profile, that is the best possible h in (1.3), is determined exactly in the form (1.12). In [29] the authors investigate quantitatively the case of…”