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.
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$\begingroup$ do you know how to do it for Riemannian manifolds? $\endgroup$– Vitali KapovitchCommented Sep 10, 2015 at 21:43
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$\begingroup$ What are the title and authors of MR3083289? I don't have access to the AMS MR database. $\endgroup$– Alex M.Commented Jun 18, 2018 at 11:41
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2 Answers
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For complete Riemannian manifolds of bounded sectional curvature the answer is YES by L.F. Tam. Exhaustion functions on complete manifolds MR2648946. Then the answer for the Hilbert metric on a properly convex manifold is also YES since such a manifold admits a Riemannian metric of bounded curvature that is bi-Lipshitz to the Hilbert metric
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I have not worked out the details but I suspect the following surface has no such function.
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.
I am really interested in properly convex manifolds with the Hilbert metric, for which I do not know the answer.