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 $I_g=\inf_{\gamma}I(\gamma)$ with $I(\gamma)=L(\gamma)(A_{in}(\gamma)^{-1}+A_{out}(\gamma)^{-1})$ where $\gamma$ is any closed simple curve in $\R^2$, $L(\gamma)$ is the length of $\gamma$, $A_{in}(\gamma)$ and $A_{out}(\gamma)$ are the area of the regions inside and outside $\gamma$ 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 $0<t<T$ when the metric $g(t)=g_{ij}(\cdot,t)=u\delta_{ij}$ is the maximal solution of the Ricci flow equation $\1 g_{ij}/\1 t=-2R_{ij}$ on $\R^2\times (0,T)$ \cite{DH} where $T>0$ is the extinction time of the solution.