# Curvature of complete conformal metrics on the open unit disk

Let $$D$$ be the unit disk in the complex plane, and assume that $$g$$ is a Riemannian metric on $$D$$ which is complete and conformal to the standard Euclidean metric. Can it be the case that the Gaussian curvature of $$g$$ approaches zero as we approach $$\partial D$$?

Yes. Take the metric with length element $$\rho(z)|dz|$$ where $$\rho(z)=(1-|z|)^{-2}$$. It is complete since $$\int^1\rho(t)dt=\infty$$, and the curvature $$-\rho^{-2}\Delta\log\rho=\rho^{-4}({\rho'}^2-\rho\rho'')=-2(1-r)^2\to 0,$$ where $$r=|z|$$ and the primes indicate differentiation with respect to $$r$$.
• @user160856: since there is no flat complete metric, there must be some such limit on the rate, but I do not see how to determine it at this moment, even for the metrics depending only on $r$. – Alexandre Eremenko Jul 13 at 11:11