It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$ admits the unique viscosity solution $u = d(\partial \Omega, - )$. Furthermore, viscosity solutions are stable in the sense that if $u_n$ is a viscosity solution to the eikonal equation, and if $u_n \Rightarrow u$ uniformly, then $u$ is again a viscosity solution to the eikonal equation.

I was wondering if similar results remain valid in a general Riemannian manifold (with reasonable assumptions like compact, without boundary)?

Specifically, I am interested in the following situation. Suppose $(M, g)$ is a compact Riemannian manifold without boundary. Let $B = B(p, \epsilon)$ be some geodesic ball around a point $p \in M$. Set $\Omega = M \setminus \overline{B}$. Then, are the following statements true?

1. $u = d(\partial \Omega, - )$ is the unique viscosity solution of the eikonal equation $\lVert \nabla u \rVert_g = 1$ inside $\Omega$, satisfying the boundary condition $u|_{\partial \Omega} = 0$.

2. If we consider the signed distance function $v$ defined as $$v(x) = d(\partial \Omega, x), \quad \text{if } x \in \Omega, \qquad \text{and} \qquad v(x) = -d(\partial \Omega, x), \quad \text{if } x \not\in \Omega,$$ then $v$ is a viscosity solution of the eikonal equation on $M$ (or possibly on some larger open set $\Omega^\prime \supset \Omega$).

3. If $u_n$ is a sequence of viscosity solutions of the eikonal equation on $\Omega$, and if $u_n \Rightarrow u$ uniformly, then $u$ is again a viscosity solution.

I was able to find this article, where Theorem 6.23 asserts that (1) above is true for a compact manifold. I would be grateful if someone could point out some references for the other points.

Any help or comment will be highly appreciated.