Does anyone know an example of a complete surface (certainly not necessarily embedded in $\Bbb R^3$) with negative curvature unbounded below? Any example I have computed so far is either not complete or has curvature bounded below. Any light that might be shed would be welcome. (Sadly, this corner of geometry is far from my expertise.)



A complete surface with $K\to -\infty$
Kazdan and Warner (MR0343206 (49 #7950) Reviewed 
Kazdan, Jerry L.; Warner, F. W.
Curvature functions for open 2-manifolds. 
Ann. of Math. (2) 99 (1974), 203–219. 
53C20 (35J05 58G15) ) 


show that a smooth function on a $\mathbb{R}^2$ is the curvature of a complete metric if and only if $\lim\_{r\rightarrow \infty} \inf\_{|p| > r} K(p) \leq 0.$ So, presumably this should give plenty of examples of the sort you seek, and maybe if you read the paper it discusses how you might construct such.


