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Isoperimetric estimates in the product of small and large volume manifolds

Let $(M^m,g)$, $(N^n,h)$ be closed Riemannian manifolds, $m,n\geq 2$, with concave isoperimetric profiles and volumes $V_M$, $V_N$ respectively. We consider a one parameter family of product manifolds of the same volume, $(X,G_λ)=(M^m\times N^n,λ^{2n}g+ λ^{-2m}h)$, $λ>0$, and estimate a lower bound for their isoperimetric profile for big $λ$. In particular, we show that for $α\in (\frac{3}{4},1)$ and $v_0 \in (0, V_MV_N)$, there is some $λ_{0}>0$, such that for $λ>λ_0$, we can bound the isoperimetric profile of $(X,G_λ)$: $$ α^{4} f_{M,λ}(v_0) \leq I_{(X,G_λ)}(v_0)\leq f_{M,λ}(v_0)$$ where $f_{M,λ}(v)= λ^{-n} V_N I_{(M,g)}(\frac{v}{V_N})$ and $ I_{(M,g)}$ is the isoperimetric profile of $(M,g)$. Moreover if $(M,g)=(S^m,g_0)$, the $m-$sphere with the round metric, in this setting, we show that some regions of the type ${ D^λ(r)\times N_λ }$, are actual isoperimetric regions in $ (S^m\times N^n,λ^{2n}g_0+ λ^{-2m}h)$ when $λ$ is big enough; being $D^λ(r)$ a disk on $(S^m,λ^{2n}g_0)$ and $N_λ=(N, λ^{-2m}h)$.