Let $(M,g)$ be a complete noncompact Riemannian manifold. Can we find an exhaustion $M=\bigcup_{i \ge 1} U_i$ such that each $U_i$ is a bounded domain with smooth boundary $\partial U_i$ which is mean convex? Here, mean convex means the mean curvature of $\partial U_i$ with respect to the outer normal vector is positive everywhere.

If not, what condition can we impose on $(M,g)$ to guarantee this? For instance, if $M$ has positive sectional curvature, then the conclusion is true.

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