dorsal/arxiv
View SchemaIsoperimetric estimates in the product of small and large volume manifolds
| Authors | Juan Miguel Ruiz, Areli Vázquez Juárez |
|---|---|
| Categories | |
| ArXiv ID | 2601.06421vv1 |
| URL | https://arxiv.org/abs/2601.06421 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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 $N_{\lambda}=(N, \lambda^{-2m}h)$.
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"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, $(X,G_{\\lambda})=(M^m\\times N^n,\\lambda^{2n}g+ \\lambda^{-2m}h)$, $\\lambda\u003e0$, 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 $\\lambda_{0}\u003e0$, such that for $\\lambda\u003e\\lambda_0$, we can bound the isoperimetric profile of $(X,G_{\\lambda})$:\n $$ \\alpha^{4} f_{M,\\lambda}(v_0) \\leq I_{(X,G_{\\lambda})}(v_0)\\leq f_{M,\\lambda}(v_0)$$\n 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 $(M,g)$.\n 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 $N_{\\lambda}=(N, \\lambda^{-2m}h)$.",
"arxiv_id": "2601.06421",
"authors": [
"Juan Miguel Ruiz",
"Areli V\u00e1zquez Ju\u00e1rez"
],
"categories": [
"math.DG"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Isoperimetric estimates in the product of small and large volume manifolds",
"url": "https://arxiv.org/abs/2601.06421",
"version": "v1"
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