It is impossible to find a conformal metric on $\mathbb{C}$ with curvature $K\leq cr^{-2}$ for all $r\geq r_0$ with some $c\gt0$ and $r_0\gt0$. We will prove this lemma using a nice lemma with a supposedly elementary proof. Then we will determine whether a simply connected smooth surface is parabolic or hyperbolic depending on growth of curvature functions providing a nice proof provided by Milnor.



References:


[Mil77]     John Milnor. On deciding whether a surface is parabolic or hyperbolic. Amer. Math. Monthly, 84(1):43-46, 1977.
Simply connected surface, Growth of curvature functions
Prerequisite:


Integral Calculus, Gauss-Green theorem
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