Isoperimetric estimates in low-dimensional Riemannian products

Let $$(T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1 \times \cdots S_{r_k}^1, dt_1^2+dt_2^2+\cdots +dt_k^2)$$ be flat tori, $$r_k\ge \ge r_2\ge r_1>0$$ and $$({\mathbb {R}}^n,g_E)$$ the Euclidean space with metric. We compute isoperimetric profile of $$(T^2\times {\mathbb {R}}^n, h_2+g_E)$$ , $$2\le n\le 5$$ for small big values volume. These computations give explicit lower bounds $$T^2\times {R}}^n$$ . also note that similar estimates $$(T^k\times h_k+g_E)$$ k\le 7-k$$ may computed, provided $$(T^{k-1}\times {R}}^{n}, h_{k-1}+g_E)$$ exist. this explicitly $$k=3$$ use symmetrization techniques product manifolds, based on work Ros (Global theory minimal surfaces (Proc. Clay Mathematics Institute Summer School, 2001). American Mathematical Society, Providence, 2005) Morgan (Ann Glob Anal Geom 30:73–79, 2006).