Minimizer of an isoperimetric ratio on a metric on $${\mathbb {R}}^2$$R2 with finite total area

Abstract Let $$g=(g_{ij})$$ g=(gij) be a complete Riemmanian metric on $${\mathbb {R}}^2$$ R2 with finite total area and let $$I_g$$ Ig be the infimum of the quotient of the length of any closed simple curve $$\gamma $$ γ in $${\mathbb {R}}^2$$ R2 and the sum of the reciprocal of the areas of the regions inside and outside $$\gamma $$ γ respectively with respect to the metric g. Under some mild growth conditions on g we prove the existence of a minimizer for $$I_g$$ Ig . As a corollary we obtain a proof for the existence of a minimizer for $$I_{g(t)}$$ Ig(t) for any $$0