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

• Main Author: Shu-Yu Hsu
• Format: Article
• Language: English
• Published: World Scientific Publishing 2018-09-01
• Series: Bulletin of Mathematical Sciences
• Subjects: Existence of minimizer; Isoperimetric ratio; Complete Riemannian metric on $${\mathbb {R}}^2$$ R 2; Finite total area
• Online Access: http://link.springer.com/article/10.1007/s13373-018-0131-3

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 $$0<t<T$$ 0<t<T when the metric $$g(t)=g_{ij}(\cdot ,t)=u\delta _{ij}$$ g(t)=gij(·,t)=uδij is the maximal solution of the Ricci flow equation $$\partial g_{ij}/\partial t=-2R_{ij}$$ ∂gij/∂t=-2Rij on $${\mathbb {R}}^2\times (0,T)$$ R2×(0,T) (Daskalopoulos and Hamilton in Commun Anal Geom 12(1):143–164, 2004) where $$T>0$$ 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).