Isoperimetric estimates in the product of small and large volume manifolds

TL;DR

This paper investigates how the isoperimetric profile of product manifolds with concave profiles behaves as a parameter varies, providing bounds and identifying isoperimetric regions for large parameter values.

Contribution

It establishes lower bounds for the isoperimetric profile of scaled product manifolds and characterizes isoperimetric regions when one factor is a sphere.

Findings

Lower bounds for isoperimetric profiles as the parameter grows

Identification of isoperimetric regions in certain product manifolds

Explicit bounds involving the isoperimetric profile of the factor 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 $N_{\lambda}=(N, \lambda^{-2m}h)$.