Isoperimetric estimates in the product of small and large volume manifolds

Abstract

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, $(X,G_{\lambda})=(M^m\times N^n,\lambda^{2n}g+ \lambda^{-2m}h)$, $\lambda>0$, and estimate a lower bound for their isoperimetric profile for big $\lambda$. In particular, we show that for $\alpha \in (\frac{3}{4},1)$ and $v_0 \in (0, V_MV_N)$, there is some $\lambda_{0}>0$, such that for $\lambda>\lambda_0$, we can bound the isoperimetric profile of $(X,G_{\lambda})$: $$ \alpha^{4} f_{M,\lambda}(v_0) \leq I_{(X,G_{\lambda})}(v_0)\leq f_{M,\lambda}(v_0)$$ where $f_{M,\lambda}(v)= \lambda^{-n} V_N I_{(M,g)}(\frac{v}{V_N})$ 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 ${ D^{\lambda}(r)\times N_{\lambda} }$, are actual isoperimetric regions in $ (S^m\times N^n,\lambda^{2n}g_0+ \lambda^{-2m}h)$ when $\lambda$ is big enough; being $D^{\lambda}(r)$ a disk on $(S^m,\lambda^{2n}g_0)$ and $N_{\lambda}=(N, \lambda^{-2m}h)$.