# Existence of smooth proper functions with bounded derivatives on manifolds

Suppose $M$ is a complete Finsler manifold of finite dimension. Is there always a smooth proper real-valued function $f$ on $M$ so that the norms of the first and second derivative are bounded ? The existence of a smooth function with bounded first derivative follows from MR3083289 applied to the function that gives distance from the base-point.

• do you know how to do it for Riemannian manifolds? – Vitali Kapovitch Sep 10 '15 at 21:43
• What are the title and authors of MR3083289? I don't have access to the AMS MR database. – Alex M. Jun 18 '18 at 11:41

The example is a smooth surface $S$ in Euclidean space with the induced Riemannian metric.
Start with the infinite circular cylinder $C$ in Euclidean space $x^2+y^2=1$. Construct S by attaching small handles near points $(x_n,y_n,z_n)=(1,0,z_n)$ on C so that $z_{n+1}=z_n + (1/n)$
The derivative of a function $f$ on $S$ must vanish at some point $p_n$ in each handle (like a Morse function). The distance between successive vanishing points is of order $1/n$. I think properness of $f$ forces the second derivative of $f$ to be unbounded.