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

TL;DR

This paper proves the existence of minimizers for an isoperimetric ratio on  with a finite area metric and applies this to Ricci flow solutions, providing a new proof for the existence of minimizers under Ricci flow.

Contribution

It establishes the existence of minimizers for an isoperimetric ratio on  with finite area metrics and applies this to Ricci flow, offering a new proof of minimizer existence during the flow.

Findings

Existence of minimizer for the isoperimetric ratio on  with finite total area.

Application of the result to Ricci flow solutions on .

New proof for the existence of minimizers under Ricci flow.

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.