Isoperimetric estimates in the product of small and large volume manifolds

Abstract

Let $(M^m,g)$, $(N^n,h)$ be closed Riemannian manifolds, $m,n\ge 2$, 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 $v_0\in (0,V_M{V}_N)$, there is some ${\lambda }_0>0$, such that for $\lambda >{\lambda }_0$, we can bound the isoperimetric profile of $(X,G_{\lambda })$: $${\alpha }^4{f}_{M,\lambda }(v_0)\le I_{(X,G_{\lambda })}(v_0)\le f_{M,\lambda }(v_0)$$ 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 $N_{\lambda }=(N,{\lambda }^{-2m}h)$.