Isoperimetric estimates in low dimensional Riemannian products

Abstract

Let $(T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1\times ...\times S_{r_k}^1,d{t}_1^2+d{t}_2^2+...+d{t}_k^2)$ be flat tori, $r_k\ge ...\ge r_2\ge r_1>0$ and $({\mathbb{R}}^n,g_E)$ the Euclidean space with 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. We compute this explicitly for $k=3$. We use symmetrization techniques for product manifolds, based on work of A. Ros and F. Morgan.