Smoothness of distance at an end of a manifold Suppose $(X,g)$ is Riemannian manifold. Let $p\in X$ be a point. The distance with respect to g
\begin{align*}
 & \text{dist}_g(p,-):X\to [0,\infty)\\
  & \text{dist}_g(p,q) = \text{inf}\{\text{length}_g(\gamma) \,|\,\gamma:[0,1]\to X, \gamma(0) = p,\gamma(1)=q,\gamma \text{ piecewise smooth}\}\,.
\end{align*}
is a continuous function, but it is in general not smooth. However, assume that $X$ is not compact. For simplicity let's assume that it has a single end. Can one choose a compact $K$, s.t. $p\in K\subset X$, and $\text{dist}_g(p,-)|_{X\backslash K}$ is smooth?
My thinking was that if I could assume that $(X,g)$ is ALE and $K$ is the set, such that $X\backslash K$ has an ALE chart on it, it should be true. But this seems rather restrictive.
 A: Every complete, connected hyperbolic surface $S$ is isometric to the quotient of the hyperbolic plane ${\mathbb H}^2$ by a discrete subgroup $\Gamma$ of isometries of ${\mathbb H}^2$ acting freely on ${\mathbb H}^2$. Given a point $z\in {\mathbb H}^2$ and $\Gamma$ as above, one defines the Dirichlet domain $D=D_{\Gamma,z}$ by
$$
\{w\in D: d(w,z)\le d(w,\gamma z) \forall \gamma\in \Gamma\}. 
$$
This domain is convex (as intersection of half-planes) and is polygonal. Topologically, one obtains $S$ by identifying boundary edges of $D$ via some elements of $\Gamma$. Thus, $D$ is noncompact if and only if $S$ is.
Now, to relate $D$ and the non-smoothness locus of the distance function on $S$, let
let $\pi: {\mathbb H}^2\to S$ be the covering map (the quotient via the $\Gamma$-action). Let
$p:=\pi(z)$. Then (if you look at the definition of the boundary of $D$), $\pi(\partial D)$ is exactly the non-smoothness locus of the function $d^2(p, \cdot)$ on $S$ (I prefer to square to avoid the nonsmoothness at $p$, otherwise, it is the same as the nonsmoothness locus of the distance function $d(p, \cdot)$): A point $q$ belongs to
$\pi(\partial D)$ precisely when there is more than one minimizing (unit speed) geodesic from $p$ to $q$. The loop formed by such distinct geodesics corresponds to an element $\gamma\in \Gamma$ such that a preimage of $q$ in ${\mathbb H}^2$ lies on the bisector of the pair $p, \gamma(p)$.
It remains to observe that $\pi(\partial D)$ is noncompact (equivalently, is unbounded in $S$) if and only if $D$ is noncompact.
Now, since you want a 1-ended example, just take a hyperbolic surface with one cusp or a 1-ended hyperbolic surface of infinite genus...
