Minimizer of an isoperimetric ratio on a metric on $\R^2$ with finite total area

Abstract

Let $g=(g_{ij})$ be a complete Riemmanian metric on $\R^2$ with finite total area and $I_g=\inf_γI(γ)$ with $I(γ)=L(γ)(A_{in}(γ)^{-1}+A_{out}(γ)^{-1})$ where $γ$ is any closed simple curve in $\R^2$, $L(γ)$ is the length of $γ$, $A_{in}(γ)$ and $A_{out}(γ)$ are the area of the regions inside and outside $γ$ respectively, with respect to the metric $g$. We prove the existence of a minimizer for $I_g$. As a corollary we obtain a new proof for the existence of a minimizer for $I_{g(t)}$ for any $0 0$ is the extinction time of the solution.