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

Question

I am trying to prove this exercise:

2. Introduce a complete Riemnnian metric on $\mathbb R^2$. Prove that \begin{equation} \lim_{r\to\infty}\left(\inf_{x^2+y^2\geq r^2}K(x,y)\right)\leq 0,\tag{1} \end{equation} 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.

Answer 1

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 $\pi/\sqrt\delta$).