Let $S$ be a regular surface homeomorphic to a sphere minus three points with $K < 0$ everywhere. How many simple and closed geodesics can there be in $S$? Prove your result.
Since $S$ is homeomorphic to a sphere minus three points, it's homeomorphic to a plane minus two points. Now, I know the following result:
$K < 0$ everywhere $\implies$ $\not\exists$ a geodesic bounding a simply connected region
So there could be:
a geodesic which bounds a region containing one point that's missing in the plane
a geodesic which bounds a region containing the other point that's missing in the plane
a geodesic which bounds a region containing both points missing in the plane
And there can't be a geodesic bounding a region wich doesn't contain both points (since then it wouldn't be missing any points and thus be simply connected), so $S$ can have at most three such geodesics. Is that it?