I'm working on problem 2 in section 5-4 of do Carmo's "Curves and Surfaces" (2nd edition), which requires us to show that if S = (x,y,f(x,y)) is a complete non-compact regular surface, then the minimum curvature if we go sufficiently far out, will be <= 0:
do Carmo's proof uses the fact that if the limit above is a positive value, say 2c, then if we go sufficiently far out we can find arbitrary large disks on which the curvature is bounded below by c, and this contradicts Bonnet's theorem which says that if such a lower bound on curvature exists then its diameter is bounded:
The problem is that Bonnet's theorem applies to complete surfaces, and the surfaces described by the arbitrary large disks don't seem to me to be complete, because e.g. I can no longer extend geodesics indefinitely; for example, I can't extend geodesics outside of said large disks. Hence we can't directly apply Bonnet's theorem here. What am I missing?
Thanks for your help!
