Minimizer of an isoperimetric ratio on a metric on {mathbb {R}}^2 with finite total area

Abstract

Let g=(g_{ij}) be a complete Riemmanian metric on {mathbb {R}}^2 with finite total area and let I_g be the infimum of the quotient of the length of any closed simple curve gamma in {mathbb {R}}^2 and the sum of the reciprocal of the areas of the regions inside and outside gamma respectively with respect to the metric g. Under some mild growth conditions on g we prove the existence of a minimizer for I_g. As a corollary we obtain a 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)=udelta _{ij} is the maximal solution of the Ricci flow equation partial g_{ij}/partial t=-2R_{ij} on {mathbb {R}}^2times (0,T) (Daskalopoulos and Hamilton in Commun Anal Geom 12(1):143–164, 2004) where T>0 is the extinction time of the solution. This existence of minimizer result is assumed and used without proof by Daskalopoulos and Hamilton (2004).