recently I encountered the following problem in my research. Roughly speaking, it asks if, on the universal covers of a closed Riemannian manifold, one can find exhaustion functions with uniformly locally controlled level sets.
Let $M^m$ be a closed $m$-dimensional Riemannian manifold of dimension at least $4.$ Suppose the universal cover $\bar{M}^m$ of $M^m$ is non-compact. Then is it possible to find a smooth function $f:\bar{M}\to[0,\infty)$ defined on $\bar{M},$ with the following properties?
A. Every sublevel set $f^{-1}([0,t])$ is compact for $t\in[0,\infty)$.
B. There exists $r,c>0,$ and a sequence of $\{t_j\}\subset [0,\infty)$ with $t_j\to\infty$ such that (i) For any point $p\in f^{-1}(t_j)$, with any $t_j$ in the sequence $\{t_j\},$ there exists a bi-Lipschitz map $\Gamma_p$ from the radius $r$ geodesic ball $B_r(p)$ centered around $p,$ to the standard unit ball $B_1^m(0)$ in $\mathbb{R}^m$, i.e.,$$\Gamma_p:B_r(p)\to B_1^m(0).$$ (ii) The map $\Gamma_p$ satisfies \begin{align} c\le\operatorname{Lip}\Gamma_p,\operatorname{Lip}\Gamma_p^{-1}\le c^{-1} ,\\ \Gamma(f^{-1}(t_j)\cap B_r(p))=B_1^{m-1}(0). \end{align} Here $\operatorname{Lip}$ is the Lipschitz constant of a map, and $B_1^{m-1}(0)$ is the $(m-1)$-dimensional unit ball of $\mathbb{R}^{m-1}$, regarded as a subset of $B_1^m(0)$, via the natural embedding of $\mathbb{R}^{m-1}$ into $$\mathbb{R}^{m-1}\times\{0\}\subset\mathbb{R}^{m-1}\times\mathbb{R}^{1}=\mathbb{R}^{m}.$$
I am aware of Shi's and Tam's results on exhaustion functions with uniform controls on the first and second derivatives. I guess that gives uniform control on the second fundamental form of level sets. However, I do not know how to proceed from their results, as I cannot rule out a scenario of a level set decomposing into several pieces in arbitrarily small radius geodesic balls.
I am sorry if this question is trivial, as my specialty is in geometric measure theory and thus I do not have enough knowledge of differential geometry on non-compact manifolds to answer the above questions. The motivation for the problem comes from solving variational problems on non-compact manifolds. Many thanks for your help and understanding!
