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

Abstract

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 $$00$$ 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).

Keywords

• Existence of minimizer
• Isoperimetric ratio
• Complete Riemannian metric on $${\mathbb {R}}^2$$ R 2
• Finite total area