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?
 A: 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.
A: The answer to your question is no.In case of surfaces the scalar curvature and Ricci curvature coincide with Gauss curvature. If an n dimensional M had a complete Riemannian metric h with nonnegative Ricci curvature then by the Bochner Weitzenbock formula  all L^2 harmonic one forms would be parallel.By an observation of Calabi M has infinite volume. Therefore all L^2 harmonic forms are zero. Now L^2 condition on one forms is a conformally invariant condition on a Riemann surface. However the disc has many nonzero L^2 harmonic one forms .This leads to a contradiction.
