Isoperimetric estimates in low dimensional Riemannian products

Let $(T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1 \times … \times S_{r_k}^1, dt_1^2+dt_2^2+…+dt_k^2)$beflattori,$r_k\geq …\geq r_2\geq r_1>0$and$(\mathbb R^n,g_E)$ the Euclidean space with the flat metric. We compute the isoperimetric profile of $(T^2\times {\mathbb{R}}^n,h_2+g_E)$, $2\le n\le 5$, for small and big values of the volume. These computations give explicit lower bounds for the isoperimetric profile of $T^2\times {\mathbb{R}}^n$. We also note that similar estimates for $(T^k\times {\mathbb{R}}^n,h_k+g_E)$, $2\le k\le 5$, $2\le n\le 7-k$, may be computed, provided estimates for $(T^{k-1}\times \mathbb R^{n+1}, h_{k-1}+g_E)$exist\mathrm{.}Wecomputethisexplicitlyfor$k=3$. We use symmetrization techniques for product manifolds, based on work of A. Ros and F. Morgan.