Smoothness of distance at an end of a manifold

Question

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.

Answer 1

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...