# Does there exists a complete Kahler metric with positive scalar curvature on Poincare disk?

Let (M,g) be a Poincare disk. Does there exists a complete Kahler metric h, such that the scalar curvature of h is positive?

Take any smooth function $f$ on the disk so that $f$ goes to infinity at the boundary and then our metric must be of the form $g=e^f(dx^2+dy^2)$. Every metric on any oriented surface is Kaehler, in this case for the usual complex structure. The curvature is the Laplacian of $f$. To get positive scalar curvature, you need to get the Laplacian to be negative. But $f$ needs to still grow to infinity as we approach the boundary, which is impossible by the maximum principle applied to concentric circles around the origin.
• Actually, if you use polar coordinates and write your $f(x^2+y^2) = h(r)$ (where $r^2 = x^2+y^2$), you'll see that the positivity of the scalar (i.e., Gauss) curvature is equivalent to $rh''(r) + h'(r) <0$ for $0<r<1$ and $h''(0)<0$ (since $h$ is an even function of $r$, we automatically have $h'(0)=0$). This easily implies that $h$ is a decreasing function of $r$ when $0<r<1$ (just integrate twice), so your metric cannot be complete. Commented Nov 11, 2016 at 10:27