On the Chern-Gauss-Bonnet integral for conformal metrics on R4

where χ(M) is the Euler number ofM . Later, Huber [Hu] extended this inequality to metrics with much weaker regularity. More importantly, he proved that such a surface M is conformally equivalent to a closed surface with finitely many punctures. The deficit in formula (1.1) has an interpretation as an isoperimetric constant. On a complete and open surface with Gaussian curvature absolutely integrable, one may represent each end conformally as R2 for some compact set K . We consider the isoperimetric ratio