# Exercise 2 of chapter IX of Do Carmo's Riemannian Geometry

I am trying to prove this exercise:

1. Introduce a complete Riemnnian metric on $$\mathbb R^2$$. Prove that $$$$\lim_{r\to\infty}\left(\inf_{x^2+y^2\geq r^2}K(x,y)\right)\leq 0,\tag{1}$$$$ where $$(x,y)\in\mathbb R^2$$ and $$K(x,y)$$ is the Gaussian curvature of the given metric at $$(x,y)$$.

Define $$A(r):=\inf_{x^2+y^2\geq r^2}K(x,y)$$. If $$r_1\leq r_2$$ we have $$A(r_1)\leq A(r_2)$$. So the limit in (1) can be $$-\infty$$ or a real number.

My attempt: Suppose, by contradction, that $$\lim_{r\to\infty}\left(\inf_{x^2+y^2\geq r^2}K(x,y)\right)>0.$$ Then there exist $$R>0$$ and $$\delta>0$$ such that $$\inf_{x^2+y^2\geq R^2}K(x,y)\geq\delta>0.$$ I would like use Bonnet-Myers's Theorem on $$N=\{(x,y)\in\mathbb R^2/x^2+y^2>R\}$$ or $$\widetilde N=\{(x,y)\in\mathbb R^2/x^2+y^2\geq R\}$$. Then we get a contradiction because $$N$$ and $$\widetilde N$$ are not compact. Since $$N$$ is not complete (because is not closed) and $$\widetilde N$$ is a manifold with boundary, I cannot use Bonnet-Myers's Theorem.

As you did, assume $$K\geq \delta$$ out of a compact set $$K$$ (e.g. $$K=\{x^2+y^2\leq R^2\}$$). Let $$p$$ be the point with coordinate $$(0, 0)$$ and $$D=\max\{d(p, x)\,|\, x\in K\}$$. Take a point $$q$$ so that $$d(p, q)=D+2\pi/\sqrt \delta$$, and $$\gamma$$ a minimal geodesic from $$p$$ to $$q$$. Then the portion of $$\gamma$$ restricted on $$[D, D+2\pi/\sqrt\delta]$$ is a piece of minimal geodesic of length $$2\pi/\sqrt\delta$$, it lies entirely in a region with $$K\geq \delta$$; according to the proof of Bonnet-Myers this is impossible (it was proved there if $$K\geq \delta$$, then geodesics cannot be minimizing beyond distance $$\pi/\sqrt\delta$$).