Large isoperimetric regions in the product of a compact manifold with Euclidean space

Abstract

Given a compact Riemannian manifold M without boundary, we show that large isoperimetric regions in M × k are tubular neighborhoods of M × {x}, with x ∈ k .

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

“…In [27] it is considered the more general product manifold given by M m × R k , where M m is an m-dimensional compact Riemannian manifold without boundary and k ≥ 1; let here N = m+k. It is proved that the isoperimetric profile h is given as in (1.12), but only the constant b is determined exactly, as…”

Isoperimetric inequalities in cylinders with density

“…In Theorem 3.3 we characterize the isoperimetric regions of large volume in a cylinder with density M × R, where M is a compact Riemannian manifold with density, the real line R is endowed with a constant density, and the product with the product density. In the Riemannian case Duzaar and Steffen [7] proved that in M × R isoperimetric sets of large volume are of the form M × [a, b], where a, b ∈ R. For higher dimensional Euclidean factors the problem was considered in [21], where the authors proved that the isoperimetric solutions of large volume in the Riemannian product M × R k are of the form M × B(x, r), where B(x, r) is an Euclidean ball, see also [11].…”